Dynamic Analysis of Planar 3-RRR Flexible Parallel Robots with Dynamic Stiffening

Abstract

In consideration of the second-order coupling quantity of the axial displacement caused by the transverse displacement of flexible beam, the first-order approximation coupling model of planar 3-RRR flexible parallel robots is presented, in which the rigid body motion constraints, elastic deformation motion constraints, and dynamic constraints of the moving platform are considered. Based on the different speed of the moving platform, numerical simulation results using the conventional zero-order approximation coupling model and the proposed firstorder approximation coupling model show that the effect of “dynamic stiffening” term on dynamic characteristics of the system is insignificant and can be neglected, and the zero-order approximation coupling model is enough precisely for catching essentially dynamic characteristics of the system. Then, the commercial software ANSYS 13.0 is used to confirm the validity of the zero-order approximation coupling model.

1. Introduction

In recent several decades, many researchers have paid more attention to the light flexible robots with high-speed, high-acceleration, and high-precision which are widely used in the assembly industry, the aerospace industry and the precision machining, and the measurement field. Essentially, the flexible parallel robots mechanism belongs to flexible multibody system. Dynamic modeling of flexible multibody system is a challenging task, in which not only rigid-flexible coupling effect must be studied but also elastic deformation coupling must be analyzed carefully.

At present, dynamic modeling and control of the flexible multibody system have received considerable attention as seen in survey papers [1–4]. Unfortunately, most of published works in this area addressed the manipulators with one flexible link. Comparing with single-link flexible manipulator, two-link flexible manipulator, or four-bar linkage flexible mechanism [5, 6], the research works on the flexible parallel robots are rather few.

Recently, few works have been done on dynamic modeling and control of complex mechanisms. Lee and Geng [7] developed a dynamic model of a flexible Stewart platform using Lagrange equations. Wang et al. [8–10] studied dynamic modeling and control of planar 3-Prismatic-joint-Revolute-joint-and-Revolute-joint (3-PRR) parallel robots. Zhang et al. [11, 12] studied dynamic modeling method and dynamic characteristics of planar 3-RRR flexible parallel robots. Q. H. Zhang and X. M. Zhang [13] also studied dynamic performance of planar 3-RRR flexible parallel robots under uniform temperature change.

For a simple flexible hub-beam system, when the system operates at a high speed, the conventional hybrid coordinate model, namely, zero-order approximation coupling model (ZOAC model) that assumes axial and transverse displacement of a point on the neutral axis of a beam are uncoupled is an invalid modeling method [14]. Kane et al. [15] found that when the nominal motion of rigid-flexible rotating beam system is very large, the deformation of the flexible system deformation exists significantly warp by the conventional dynamic model and experimental method, and he thought the premature linearization would lead to lack of “dynamic stiffening” term and firstly proposed “dynamic stiffening” phenomenon. Since then, many scholars start to study the so-called “dynamic stiffening” terms of rigid-flexible coupling system and different methods are used to capture it [14, 16–18]. Mayo et al. [16] reviewed different formulations to account for the geometric stiffening or dynamic stiffening effect arising from deflections large enough to cause significant changes in the configuration of the system. Lugrís et al. [17] used two different methods to calculate dynamic stiffening of the flexible system: one method is from the deformation energy and the other is from foreshortening. Zhang et al. [18] indicated that when the rotational frequency of slender beams meets or exceeds the fundamental frequency, the elastic deformation of flexible beam that is obtained by the conventional ZOAC model tends to diverge. Yu and Hong [19] summarized several methodologies for analyzing dynamic stiffening of flexible system. In consideration of structural geometric nonlinear effects, Wu and Haug [20] established a substructure synthesis formulation of flexible multibody system. Yang [21] studies dynamic model of rigid-flexible system from both theory and experiment. In consideration of dynamic stiffening effect, Liu and Lu [22] established the rigid-flexible coupling dynamic equations of three-dimensional hub-beams system using velocity variational principle and finite element method.

Since dynamic stiffening phenomenon was proposed by Kane et al., many relevant literatures were published [14, 16–22], but they mainly focus on the flexible rotation beam system. To my knowledge, dynamic stiffening phenomenon of flexible parallel robots has never been studied. Because dynamic stiffening term was not neglected in the high-speed flexible rotation beam system, we do not know that how dynamic stiffening of flexible parallel robots influences its owe dynamic characteristics. So, it is important for us to study dynamic stiffening of flexible parallel robots.

The remainder of the paper is organized as follows. Section 2 introduces dynamic modeling of planar 3-RRR flexible parallel robots with dynamic stiffening. In consideration of the second-order coupling quantity of the axial displacement caused by the transverse displacement of flexible beam, based on the finite element method (FEM) and the Lagrange equation, the first-order approximation coupling model (FOAC model) of planar 3-RRR flexible parallel robots is presented. Section 3 discusses the constraint equations of planar 3-RRR flexible parallel robots system which include rigid body motion constraints, elastic deformation motion constraints, and dynamic constraints of the moving platform. In Section 4, numerical results are presented based on the ZOAC model and FOAC model. One finds that the influence of dynamic stiffening of planar 3-RRR flexible parallel robots on dynamic characteristics can be neglected, and the ZOAC model can effectively reflects dynamic characteristics of the system (Figure 7). The commercial ANSYS 13.0 software is used to confirm the validity of the ZOAC model. Finally, the conclusions are given in Section 5.

2. Dynamic Modeling of Planar 3-RRR Flexible Parallel Robots with Dynamic Stiffening

The flexible link can be modeled by connecting a series of beam elements. Figure 1 shows a beam element before and after deformation. The 𝑂-𝑋𝑌 is the global fixed frame and the A-xy is the local moving frame with Ax axis coincident with the neutral line of the beam element. Its original point A is located at one node of the beam element before deformation. B is another node of the beam element. The 𝑂-𝑥^′𝑦^′ system is an intermediate coordinate frame whose origin is rigidly attached to the origin of the 𝑂-𝑋𝑌 and whose axes are parallel to the axes of the local moving frame A-xy. 𝜑 is the angle between the global fixed frame 𝑂-𝑋𝑌 and the intermediate coordinate frame 𝑂-𝑥^′𝑦^′. Considering the general point 𝐶 in the beam element, let point 𝐶₀ be the corresponding point on the neutral line. Points 𝐶^′ and 𝐶^′₀ are their positions after deformation, respectively. The elastic deformation of the point 𝐶₀ in the 𝐴-𝑥𝑦 is given by 𝐯₀()=𝑣𝑥,𝑡¹⁰𝑣₂₀=𝐍(𝑥)𝐞_𝑓, (1) where 𝐞_𝑓=𝑒₁𝑒₂𝑒₃𝑒₄𝑒₅𝑒₆𝑒₇𝑒₈^𝑇 is the nodal displacement vector, in which 𝑒₁ and 𝑒₅ are the axial displacements of two nodes 𝐴 and 𝐵, respectively; 𝑒₂ and 𝑒₆ are the transverse displacements; 𝑒₃ and 𝑒₇ are elastic rotational angles; 𝑒₄ and 𝑒₈ are section curvatures; and the superscript 𝑇 indicates matrix transpose. 𝐍(𝑥) is the shape function matrix. Assuming that the axial displacement 𝐯₁₀ of point 𝐶₀ is a linear function, the transverse displacement 𝐯₂₀ is a fifth-order hermit function, yielding 𝐍=𝐍₁(𝑥)𝐍₂(𝑥)=𝑓₁000𝑓₅0000𝑓₂𝑓₃𝑓₄0𝑓₆𝑓₇𝑓₈, (2) where 𝑓₁𝑓=1−𝜉,₂=1−10𝜉³+15𝜉⁴−6𝜉⁵,𝑓₃=𝑙𝜉−6𝜉³+8𝜉⁴−3𝜉⁵,𝑓₅𝑓=𝜉,₄=𝑙²𝜉²−3𝜉³+3𝜉⁴−𝜉⁵2,𝑓₆=10𝜉³−15𝜉⁴+6𝜉⁵,𝑥𝜉=𝑙,𝑓₇=𝑙−4𝜉³+7𝜉⁴−3𝜉⁵,𝑓₈=𝑙²𝜉³−2𝜉⁴+𝜉⁵2. (3)𝑙 is length of the beam element.

Figure 1 

Beam element deformation.

The deformation displacements of 𝐶^′ in the A-xy can be written as [14] 𝑣𝐯=¹()𝑣𝑥,𝑡₂()=𝑣𝑥,𝑡¹⁰−𝑦𝜕𝑣₂₀𝜕𝑥−^𝑥₀12𝜕𝑣₂₀𝜕𝜉²𝑣𝑑𝜉₂₀; (4) then, the displacement of 𝐶^′ can be expressed in the global fixed frame by 𝐫_𝑒,𝐶′=𝐫_𝑒,𝐴+𝐑𝐞₀+𝐯, (5) where 𝐑 is the planar transformation matrix defined as 𝐑=cos𝜑−sin𝜑sin𝜑cos𝜑; (6)𝐞₀ is the location coordinates vector of the point 𝐶 in the A-xy system and is given by 𝑥𝑦^𝑇.

Taking the first derivative on (5) and considering (1), the velocity vector of the point 𝐶 in the 𝑂-𝑋𝑌 system can be written as 𝐫̇_𝑒,𝐶′=𝐫̇_𝑒,𝐴+𝐑_𝜑𝐞₀+𝐍𝐞_𝑓+𝑤_𝑐0𝜑̇𝐞+𝐑𝐍−^𝑇𝑓𝐇_𝑒𝟎𝐞̇_𝑓=𝐈𝐑_𝜑𝐞₀+𝐍𝐞_𝑓+𝑤_𝑐0𝐑𝐞𝐍−^𝑇𝑓𝐇_𝑒𝟎×𝐫̇_𝑒,𝐴𝜑̇𝐞̇_𝑓=𝐒𝐮, (7) where 𝐒=𝐈𝐑_𝜑𝐞₀+𝐍𝐞_𝑓+𝑤_𝑐0𝐑𝐞𝐍−^𝑇𝑓𝐇_𝑒𝟎,𝐮=𝐫^𝑇_𝑒,𝐴𝜑𝐞^𝑇_𝑓^𝑇, (8)𝐫̇_𝑒,𝐴 is the velocity vector of the original point A in the 𝑂-𝑋𝑌 system, 𝐇_𝑒=^𝑥₀𝜕𝐍₂𝜕𝜕𝜉𝐍^𝑇₂𝐑𝜕𝜉𝑑𝜉,_𝜑=𝜕𝐑𝜕𝜑,𝐍=𝐍^𝑇₁𝜕−𝑦𝐍^𝑇₂𝜕𝑥𝐍^𝑇₂, (9)𝐈 is 2 × 2 unit matrix, and parameter 𝑤_𝑐=−(1/2)𝐞^𝑇_𝑓𝐇_𝑒𝐞_𝑓 is the second-order coupling term that is the axial shrinking quantity caused by the transverse displacement 𝑣₂. In the ZOAC model, the small deformation assumption is adopted, so 𝑣₁(𝑥,𝑡)=𝑣₁₀−𝑦(𝜕𝑣₂₀/𝜕𝑥) is taken, and 𝑤_𝑐 is neglected.

2.1. Kinetic Energy of the Beam Element

Using (7), kinetic energy of the beam element which includes translational kinetic energy of the beam element, rotational kinetic energy of the cross-section, and kinetic energy of lumped mass at the end points of beam can be written as 𝑇_𝑒=12_𝑉𝜌𝐫̇^𝑇_𝑒,𝐶′𝐫̇_𝑒,𝐶′+1𝑑𝑉2^𝑙₀𝜃̇²(𝑥,𝑡)𝑑𝐽_𝑐+12𝑚_𝐿𝐫̇^𝑇_𝐴′𝐫̇_𝐴′+𝑚_𝑅𝐫̇^𝑇_𝐵′𝐫̇_𝐵′+12𝐽_𝐿𝜃̇²(0,𝑡)+𝐽_𝑅𝜃̇²(=1𝑙,𝑡)2𝐮̇^𝑇𝐦_𝑒𝐮̇, (10) where 𝐦_𝑒=𝐦_𝑒𝑟𝑟𝐦_e𝑟𝜑𝐦_𝑒𝑟𝑓𝑚_𝑒𝜑𝜑𝐦_𝑒𝜑𝑓Symmetry𝐦_𝑒𝑓𝑓,𝐦_𝑒𝑟𝑟=_𝑉𝜌𝑑𝑉+𝑚_𝐿+𝑚_𝑅𝐦𝐈,_𝑒𝑟𝜑=𝐑_𝜑𝐍₁+𝐑_𝜑𝐍₂𝐞_𝑓+𝐑_𝜑𝑉𝜌𝑤_𝑐0𝑑𝑉+𝐑_𝜑𝐍_𝑐𝐦𝐞,_𝑒𝑟𝑓=𝐑𝐍₂−𝐑_𝑉𝜌𝐞^𝑇_𝑓𝐇_𝑒𝟎𝑑𝑉+𝐑𝐍_𝑐,𝑚_𝑒𝜑𝜑=𝐍₃+2𝐍₄𝐞_𝑓+𝐞^𝑇_𝑓𝐍₈𝐞_𝑓+𝐞^𝑇_𝑓𝐍₅𝐞_𝑓+𝜌𝐼𝑙+𝐞^𝑇𝐍_𝑐𝐞+𝐽_𝐿+𝐽_𝑅,𝐦_𝑒𝜑𝑓=𝐍₆+𝐞^𝑇_𝑓𝐍₇+𝐞^𝑇_𝑓𝐍₉+𝐞^𝑇𝐍_𝑐+𝐂₁,𝐦_𝑒𝑓𝑓=𝐍₅+𝐂₂,𝐍₁=_𝑉𝜌𝐞₀𝐍𝑑𝑉,₂=_𝑉𝜌𝐍𝑑𝑉,𝐍₃=_𝑉𝜌𝐞^𝑇₀𝐞₀𝐍𝑑𝑉,₄=_𝑉𝜌𝐞^𝑇₀𝐍𝐍𝑑𝑉,₅=_𝑉𝜌𝐍^𝑇𝐍𝑑𝑉,𝐍₆=_𝑉𝜌𝐞^𝑇₀̃𝐍𝐈𝐍𝑑𝑉,₇=_𝑉𝜌𝐍^𝑇̃𝐍𝐈𝐍𝑑𝑉,₈=−_𝑉𝜌𝑥𝐇_𝑒𝑑𝑉,𝐍₉=_𝑉𝜌𝑦𝐇_𝑒𝑑𝑉,𝐍_𝑐=𝑚_𝐿𝐍(0)+𝑚_𝑅𝐍(𝑙),𝐍_𝑐=𝑚_𝐿𝐍(0)^𝑇𝐍(0)+𝑚_𝑅𝐍(𝑙)^𝑇𝐍(𝑙),𝐍_𝑐=𝑚_𝐿𝐍(0)^𝑇𝐈𝐍̃(0)+𝑚_𝑅𝐍(𝑙)^𝑇̃(𝑙),𝐂𝐈𝐍₁=^𝑙₀𝜌𝐼𝐍^′₂^𝑇𝑑𝑥+𝐽_𝐿𝐍^′₂^𝑇(0)+𝐽_𝑅𝐍^′₂^𝑇(𝑙),𝐂₂=^𝑙₀𝜌𝐼𝐍^′₂𝐍^′₂^𝑇𝑑𝑥+𝐍_𝑐+𝐽_𝐿𝐍^′₂(0)𝐍^′₂^𝑇(0)+𝐽_𝑅𝐍^′₂(𝑙)𝐍^′₂^𝑇(𝑙). (11)𝐈̃ is skew symmetric matrix, 𝜃(𝑥,𝑡)=𝜑+𝜕𝑣₂(𝑥,𝑡)/𝜕𝑥 is absolute rotation angle of microsegment 𝑑𝑥 which distances 𝑥 from the original point A. 𝑚_𝐿, 𝐽_𝐿, 𝑚_𝑅, and 𝐽_𝑅 are lumped mass and lumped moment of inertia at left and right end points of the beam element, respectively. 𝐍^′₂=𝜕𝐍₂/𝜕𝑥. It should be mentioned that the high-order terms related with 𝑤_𝑐, such as 𝑣₁₀𝑤_𝑐, 𝑤̇²_𝑐, 𝑣₂𝑤̇_𝑐, and 𝑣̇₂𝑤_𝑐, are omitted since 𝑤_𝑐, 𝑤̇_𝑐, 𝑣₁₀, 𝑣₂, and 𝑣̇₂ are very small.

2.2. Strain Energy of the Beam Element

Nonlinear terms in the strain-displacement relationship are neglected [16]. Then, strain energy of the beam element can be written as 𝑈_𝑒=12^𝑙₀𝐸𝐼(𝑥)𝑣^′′₂₀(𝑥,𝑡)²+1𝑑𝑥2^𝑙₀𝐸𝑆(𝑥)𝑣^′₁₀(𝑥,𝑡)²=1𝑑𝑥2𝐮^𝑇𝐤_𝑒𝐮, (12) where 𝑣^′₁₀(𝑥,𝑡)=𝜕𝑣₁₀(𝑥,𝑡)/𝜕𝑥, 𝑣^′′₂₀(𝑥,𝑡)=𝜕²𝑣₂₀(𝑥,𝑡)/𝜕𝑥², and 𝐸 is elastic modulus of materials. 𝐼(𝑥),𝑆(𝑥) are cross-sectional moment of inertia and cross-sectional area of the beam element, respectively. Consider𝐤_𝐞=𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝐤_𝑒𝑓𝑓,𝐤_𝑒𝑓𝑓=^𝑙₀𝐸𝐼(𝑥)𝐍^′′₂^𝑇(𝑥)𝐍^′′₂(𝑥)+𝑑𝑥_𝑙0𝐸𝑆(𝑥)𝐍^′₁^𝑇(𝑥)𝐍^′₁(𝑥)𝑑𝑥. (13)

2.3. Dynamic Equation of the Beam Element

According to Lagrange’s equation, dynamic equation of the beam element can be written as 𝐦_𝑒𝐮̈+𝐤_𝑒𝐮=𝐩_𝑒,𝑒+𝐩_𝑒,𝑣, (14) where 𝐩_𝑒,𝑒 and 𝐩_𝑒,𝑣 are the generalized external forces and the quadratic velocity vector that contains the gyroscopic and the Coriolis force components, respectively.

The elemental dynamic equation has been established in the A-xy system. Before forming the dynamic equation of the system, (14) must be expressed in the 𝑂-𝑋𝑌 system. Define the coordinate transformation matrix 𝐁 as follows: 𝐁=cos𝜑sin𝜑000000−sin𝜑cos𝜑00000000100000000100000000cos𝜑sin𝜑000000−sin𝜑cos𝜑000000001000000001. (15) Let 𝐞_𝑓=𝐁𝐄_𝑒, (16) where 𝐄_𝑒 is the elemental nodal coordinate vector in the 𝑂-𝑋𝑌 system. Taking the first and the second derivative on (16) with respect to time yields 𝐞̇_𝑓=𝐁_𝜑𝐄_𝑒𝜑̇+𝐵𝐄̇_𝑒,𝐞̈_𝑓=𝐁_𝜑𝜑𝐄_𝑒𝜑̇²+2𝐁_𝜑𝐄̇_𝑒𝜑̇+𝐁_𝜑𝐄_𝑒𝜑̈+𝐁𝐄̈_𝑒, (17) where 𝐁_𝜑=𝜕𝐁/𝜕𝜑, 𝐁_𝜑𝜑=𝜕²𝐁/𝜕𝜑². Substitute(16) and (17) into (14) and premultiply by matrix 𝐈𝟎𝟎𝟎1𝐄^𝑇_𝑒𝐁_𝜑𝟎𝟎𝐁^𝑇. (18) Then, dynamic equation of the beam element can be expressed in the 𝑂-𝑋𝑌 system by 𝐌_𝑒𝐔̈_𝑒+𝐊_𝑒𝐔_𝑒=𝐐_𝑒,𝑒+𝐐_𝑒,𝑣, (19) where 𝐔_𝑒=𝐫^𝑇_𝑒,𝐴𝜑𝐄^𝑇_𝑒^𝑇, (20) where 𝐌_𝑒 and 𝐊_𝑒 are 8×8 the element generalized mass and stiffness matrixes in the 𝑂-𝑋𝑌 system, respectively. 𝐐_𝑒,𝑣 is 8 × 1 the quadratic velocity vector in the 𝑂-𝑋𝑌 system. 𝐐_𝑒,𝑒 is 8×1 the generalized external forces vector in the 𝑂-𝑋𝑌 system.

3. Governing Dynamic Equation of Planar 3-RRR Flexible Parallel Robots

In this section, according to the above dynamic equation of the beam element, governing dynamic equation of the planar 3-RRR parallel robots can be established, in which constraint equations are considered.

3.1. The Sketch of Planar 3-RRR Parallel Robots

The sketch of planar flexible 3-RRR parallel robots is constructed by the regular triangle moving platform 𝐶₁𝐶₂𝐶₃, the static platform, and three symmetrical kinematic chains 𝐴₁𝐵₁𝐶₁, 𝐴₂𝐵₂𝐶₂, and 𝐴₃𝐵₃𝐶₃, as shown in Figure 2. Each kinematic chain has one active revolute (R) joint followed by two consecutive passive revolute (R) joints. The active revolute joints are installed at 𝐴_𝑖, 𝑖=1,2,3. 𝐴₁,  𝐴₂ and𝐴₃ are the regular triangle's three vertices. 𝐴₁𝐵₁=𝐴₂𝐵₂=𝐴₃𝐵₃, and 𝐵₁𝐶₁=𝐵₂𝐶₂=𝐵₃𝐶₃. The vertexes 𝑂 and 𝑃 are centers of the regular triangles 𝐴₁𝐴₂𝐴₃ and 𝐶₁𝐶₂𝐶₃, respectively. The 𝑂-𝑋𝑌 is the global fixed frame. Parameters 𝛼_𝑖, 𝛽_𝑖(𝑖=1,2,3) are the angles between the 𝑋-axis of the fixed frame and linkages 𝐴_𝑖𝐵_𝑖, 𝐵_𝑖𝐶_𝑖(𝑖=1,2,3), respectively. 𝜃 is the angle between the 𝑋-axis of the fixed frame and side 𝐶₁𝐶₂ of the regular triangle 𝐶₁𝐶₂𝐶₃. 𝐿₁, 𝐿₂,𝐿₃, and 𝐿₄ are the length of the segments 𝐴₂𝐵₂, 𝐵₂𝐶₂, 𝐶₂𝑃, and 𝑂𝐴₂.

Figure 2 

The sketch of planar 3-RRR parallel robot.

3.2. Constraint Equations

Different from the simple structure, constraint relationships of planar 3-RRR of flexible parallel robots which include the rigid body motion constraints, the elastic deformation motion constraints, and dynamic constraints of the moving platform are very complex and can be used to eliminate the correlation of generalized coordinates.

3.2.1. Constraint Equations of the Rigid-Body Motion

Generalized coordinates are formed by rigid-body motion coordinates and elastic coordinates. As shown in Figure 2, rigid-body motion coordinates that include drive joint rotation angles 𝜶=𝛼₁𝛼₂𝛼₃^𝑇, passive joint rotation angles 𝜷=𝛽₁𝛽₂𝛽₃^𝑇, translation displacement and rotation angle of the moving platform 𝐗_𝑃=𝑋_𝑃𝑌_𝑃𝜃^𝑇, and rigid body motion coordinate vector 𝜶, 𝜷, 𝐗_𝑃 are not independent. Because three kinematic chains of 3-RRR parallel robots are full symmetrical, one chain is studied. As shown in Figure 3, a closed-loop vector equation can be established 𝑂𝐴_𝑖+𝐴_𝑖𝐵_𝑖+𝐵_𝑖𝐶_𝑖+𝐶_𝑖𝑃−𝑂𝑃=0. (21) Projecting (21) into 𝑋 and 𝑌 component yields 𝑋_𝐴𝑖+𝐿₁cos𝛼_𝑖+𝐿₂cos𝛽_𝑖+𝐿₃𝜋cos6+2(𝑖−1)𝜋3+𝜃−𝑋_𝑃𝑌=0,_𝐴𝑖+𝐿₁sin𝛼_𝑖+𝐿₂sin𝛽_𝑖+𝐿₃𝜋sin6+2(𝑖−1)𝜋3+𝜃−𝑌_𝑃=0,𝑖=1,2,3, (22) where (𝑋_𝐴𝑖,𝑌_𝐴𝑖) and (𝑋_𝑃,𝑌_𝑃) are the coordinates of the points 𝐴_𝑖 and 𝑃 in the 𝑂-𝑋𝑌 system, respectively.

Figure 3 

The flexible kinematic chain.

3.2.2. Constraint Equations of the Elastic Deformation Motion

As shown in Figure 3, center point 𝑃 and vertex 𝐶_𝑖 of the moving platform are moved to the points 𝑃^′,𝐶^′_𝑖 because of elastic deformation motion of the flexible links 𝐴_𝑖𝐵_𝑖, 𝐵_𝑖𝐶_𝑖(𝑖=1,2,3). Setting P-xy is the local coordinate system with original point P, and 𝑃^′-𝑥^′𝑦^′ is the elastic coordinate system with original point 𝑃^′. 𝐓,𝐓 and 𝐓 are the coordinate transformation matrixes for 𝑝-𝑥𝑦→𝑂-𝑋𝑌,𝑝^′-𝑥^′𝑦^′→𝑝-𝑥𝑦 and 𝑝^′-𝑥^′𝑦^′→𝑂-𝑋𝑌, respectively. Then, 𝐓=𝐓𝐓. 𝑈_𝑖22,𝑈_𝑖23,𝑈_𝑖24 are elastic displacements and elastic rotational angle at the end point 𝐶_𝑖 of the flexile link 𝐵_𝑖𝐶_𝑖, and the section curvature is equal to zero at the point 𝐶_𝑖. Δ𝑋_𝑃, and Δ𝑌_𝑃,𝜀 are translational and rotational declinations of the moving platform because of elastic vibration of the flexible links. Assuming that the moving platform is rigid, so 𝑃𝐶_𝑖=𝑃^′𝐶^′_𝑖. Noting that 𝐓=cos𝜃−sin𝜃𝑋_𝑃sin𝜃cos𝜃𝑌_𝑃,001𝐓=1−𝜀Δ𝑋_𝑃𝜀1Δ𝑌_𝑃001. (23) Assuming that 𝑋_𝐶𝑖𝑌_𝐶𝑖 and 𝑋_𝐶′𝑖𝑌_𝐶′𝑖 are the coordinates of the points 𝐶_𝑖 and 𝐶^′_𝑖 in the 𝑂-𝑋𝑌 system yields 𝑋_𝐶′𝑖𝑌_𝐶′𝑖1=𝐓𝑥^′_𝐶′𝑖𝑦^′_𝐶′𝑖1_𝑃′,𝑋_𝐶𝑖𝑌_𝐶𝑖1𝑥=𝐓𝑦^𝐶𝑖_𝐶𝑖1_𝑃, (24) where (𝑥_𝐶𝑖,𝑦_𝐶𝑖)_𝑝=(𝑥^′_𝐶′𝑖,𝑦^′_𝐶′𝑖)_𝑝′=(𝐿₃cos(𝜋/2+2𝑖𝜋/3),𝐿₃sin(𝜋/2+2𝑖𝜋/3)). ()_𝑝 means the description in the p-xy.

Setting 𝐓₁=cos𝜃−sin𝜃sin𝜃cos𝜃 (25) yields 𝑈_𝑖22𝑈_𝑖23=𝑋_𝐶′𝑖𝑌_𝐶′𝑖−𝑋_𝐶𝑖𝑌_𝐶𝑖=𝐈𝐓₁−𝑦_𝐶𝑖𝑥_𝐶𝑖𝑃Δ𝑋_𝑃Δ𝑌_𝑃𝜀𝜀=³_𝑖=1𝑈_𝑖24, (26) where 𝐈 is 2×2 unit matrix.

3.2.3. Dynamic Constraints of the Moving Platform

Assuming that 𝐅_𝑖 is the generalized joint constraint antiforce that the passive joint 𝐶_𝑖 are applied to the moving platform. 𝑀_𝑣, 𝐽_𝑣 are the mass and moment of inertia of the moving platform. Then, dynamic constraint of the moving platform can be expressed by 𝑀_𝑣𝑀_𝑣𝐽_𝑣Δ𝑋̈_𝑃Δ𝑌̈_𝑃𝜀̈=³_𝑖=1𝐅_𝑖−𝑀_𝑣𝑋̈_𝑃𝑀_𝑣𝑌̈_𝑃𝐽_𝑣𝜀̈. (27)

4. Governing Dynamic Equation of the Flexible Robots System

Considering constraint equations ((22)–(27)) and assembling all the element dynamic equations with respect to the compatibility at the nodes, then the governing dynamic equation that describes the dynamic characteristic of system can be formed as 𝐌𝐔̈+𝐊𝐔=𝐐_𝑒+𝐐_𝑣, (28) where 𝐔 is the generalized coordinate vector of planer 3-RRR flexible parallel robots in the 𝑂-𝑋𝑌 system which includes rigid-body motion coordinates, node elastic deformation coordinates, and the declinations of the moving platform. Equation (28) is the nonlinear and rigid-flexible coupling differential equation and can be decomposed into the rigid subsection and the flexible subsection as 𝐌_𝑟(𝐔)𝜶̈=𝐐^𝑟_𝑣𝐔̇𝐌𝐔,+𝝉,_𝑓(𝜶)𝐔̈+𝐂_𝑓(𝜶,𝜶̇)𝐔̇+𝐊_𝑓(𝜶,𝜶̇,𝜶̈)𝐔=𝐐_𝑓(𝜶,𝜶̇,𝜶̈). (29)(30) Now, one will emphatically study flexible subsection (30), in which generalized stiffness matrix 𝐊_𝑓(𝜶,𝜶̇,𝜶̈) can be expressed by 𝐊(𝜶,𝜶̇,𝜶̈)=_𝑒𝐃^𝑇_𝑒𝐁^𝑇𝐍₅+𝐂₂𝐁_𝜑𝜑−2𝐍₇+𝐍_𝑐𝐁_𝜑−𝐍₅+𝐍_𝑐𝐁𝜑̇²+𝐍₅+𝐂₂𝐁_𝜑+𝐍^𝑇₇+𝐍^𝑇_𝑐𝐁𝜑̈+𝐤_𝑓𝑓𝐁𝐃_𝑒+_𝑒𝐃^𝑇_𝑒𝐁^𝑇𝑔_1𝑒𝐇_𝑒+𝐍^𝑇₉𝜑̈−𝐍₈𝜑̇²𝐁𝐃_𝑒=𝐊₁+𝐊_𝑑, (31) where 𝐇_𝑒=∫_𝑉−𝜌𝐇^𝑇_𝑒𝑑𝑉, 𝐑^𝑇𝐫̈_𝑒,𝐴=^𝑔1𝑒_𝑔2𝑒, 𝐃_𝑒 is the coordinate compatibility matrix, ∑_𝑒 means summing to all elements, and 𝜑 is a function of rigid body motion variable 𝜶. The underline term is the dynamic stiffening term [14]. In the conventional ZOAC model, the term is equal to zero. In general, when the system operates at a high-speed and high-acceleration such as the flexible hub-beam [14–22], the term cannot be neglected.

5. Numerical Simulation

In this section, dynamic responses of planar 3-RRR flexible parallel robots are studied using the ZOAC model and FOAC model, comparing influences of the dynamic stiffening term on dynamic response under the different speeds of the moving platform.

Assuming that all links that include three active links and three passive links are flexible and that other parts are rigid, every flexible link is divided into three equal length beam elements. Assume that the material of 3-RRR system is the aluminum alloy, the thickness of the moving platform 𝑐=0.0034m, 𝐿₁=0.245m, 𝐿₂=0.242m,  𝐿₃=0.108m, 𝐿₄=0.4m, the lumped mass of the joint 𝑀_𝑣=0.02kg, and the lumped moment of inertia 𝐽_𝑣=0.00005kgm². The other parameters of flexible links are defined as in Table 1.

Table 1 

Parameters of the flexible links of 3-RRR.

It is assumed that the trajectory of the moving platform is described as 𝑋_𝑃()𝑌=0.04cos𝑤𝜋𝑡−0.02,_𝑃(),𝜋=0.04sin𝑤𝜋𝑡𝜃=4, (32) where 𝑤 is angular frequency that can reflect moving speed of the motion platform. Due to the requirement of following numerical simulation, the parameter 𝑤 is chosen to be 10𝜋rad/s and 350𝜋rad/s, respectively. Let the moving platform continuously operate four motion cycles along with the given circle trajectory Equation (32). Firstly, considering the parameter 𝑤=10𝜋rad/s in (32), in this case, the acceleration of the moving platform is 4g (g is weight acceleration), and the maximum absolute angle speed of the drive joints 𝑤_𝑖, 𝑖=1,2,3 reaches to 5.46, 5.19, and 5.2 rad/s, respectively. Figure 4 shows elastic displacements and rotation angle of the moving platform with respect to four motion cycles based on the ZOAC model and the FOAC model when 𝑤=10𝜋 rad/s, respectively. Figure 5 shows elastic displacement responses which are located at the 1/3 length of active link 1. In all figures, the slide line with ∘ is a result using the ZOAC model, the slide line with + is a result using the FOAC model, and the slide line with ∗ is a result using ANSYS simulation. Figure 6(a) shows numerical result of the maximum stress response of flexible links; meanwhile, the simplified 3D model is developed by SolidWorks software and then imported into Workbench environment which is a simulation module of ANSYS 13.0 software for transient dynamic analysis as shown in Figure 6(b). All curves show periodical change in Figures 4–6.

(a)

(b)

(c)

Figure 4 

(a) and (b) are elastic displacements of the moving platform along with 𝑋, 𝑌 direction, respectively; (c) is elastic rotational angle of the moving platform in 𝑋𝑌 planar, when 𝑤=10𝜋 rad/s.

(a)

(b)

Figure 5 

(a) and (b) are elastic displacements of along with 𝑋, 𝑌 direction which are located at the 1/3 length of the active link 1, respectively, when 𝑤=10𝜋 rad/s.

(a)

(b)

Figure 6 

(a) is maximum stress response of all flexible links through numerical calculation and (b) is ANSYS results, when 𝑤=10𝜋 rad/s.

(a)

(b)

(c)

(d)

Figure 7 

(a), (b), (c), and (d) are the first four-order modal shapes of the system, respectively.

Form Figures 4, 5, and 6(a), we can find that the numerical results are very close using the ZOAC model and the FOAC model. The influence of dynamic stiffening on dynamic characteristic of the system can be neglected when 𝑤=10𝜋 rad/s. Reviewing Figures 6(a) and 6(b), one can find that numerical results are similar with ANSYS simulation results, and the change law of the curves are consistent and show periodic variation. Then, the validity of ZOAC model and FOAC model is verified.

To illustrate the validity of the theory model, modal analysis module of Workbench is used to calculate natural frequencies of the system. Theory results using the theory model and ANSYS result using ANSYS-Workbench software are listed in Table 2. From Table 2, we can find that the difference of the first natural frequency is 0.99% and very small, and the difference of the second natural frequency is 8.91%, the difference of the third natural frequency is big and reaches to 37.14%, the fourth-order is 11.74%, meanwhile, all the first three-order natural frequencies are smaller than the fourth-order natural frequency using two methods. The existence of difference can be explained as follows: (1) all components are flexible in the ANSYS simulation, but, just only the links are flexible in theory model, (2) Meshing is different, and (3) solver may be different. Form the difference toleration and the consistent of the natural frequency change, ANSYS result can be used to verify the validity of the theory model.

Table 2 

The first four-order natural frequencies of the system (Hz).

To further verify, influence of stiffening dynamic of 3-RRR flexible parallel robots is slight and can be neglected in the mathematics model. Let 𝑤=350𝜋 rad/s in (32). Then, the acceleration of the moving platform is 4836 g (Unbelievable), and the maximum absolute angle speed of the drive joints 𝑤_𝑖, 𝑖=1,2,3 reaches to 191.02 (30.04 Hz), 181.77 (28.93 Hz), and 182.58 rad/s (29.06 Hz) and exceeds the first natural frequency 27.177 Hz (or 27.446 Hz), as shown in Figures 8 and 9. Figures 8(a), 8(b), and 8(c) are elastic displacements and rotation angle of the moving platform with respect to four motion cycles using the ZOAC model and the FOAC model when 𝑤=350𝜋 rad/s, respectively. Figure 8(d) is the maximum stress response of flexible links with respect to four motion cycles. Figures 9(a) and 9(b) are elastic displacement responses along with 𝑋- axisand𝑌- axis which are located at the 1/3 length of active link 1, respectively.

(a)

(b)

(c)

(d)

Figure 8 

(a) and (b) are elastic displacements of the moving platform along with 𝑋, 𝑌 direction, respectively, (c) is elastic rotational angle of the moving platform in 𝑋𝑌 planar, and (d) is maximum stress response of all flexible links through numerical calculation, when 𝑤=350𝜋 rad/s.

(a)

(b)

Figure 9 

: (a) and (b) are elastic displacements of along with X,Y direction which are located at the 1/3 length of the active link 1, respectively, when 𝑤=350𝜋 rad/s.

From Figures 8 and 9, we find that although the acceleration of the moving platform has reached to 4836 g m/s² and the angle speed of the drive joints has exceeded the first natural frequency of the system, all curves using the ZOAC model are still very close to the relevant curves using the FOAC model. The influence of dynamic stiffening is still very slight. At the moment, elastic displacements and rotational angle of the moving platform and flexible link 1 increase two orders of magnitude more than 𝑤=10𝜋 rad/s. The maximum stress of flexible links changes from 67.99 MPa to 125600 MPa and increases four orders of magnitude. When increasing 𝑤 in (32) again, there are just only increasing vibration amplitude of flexible links and the moving platform and maximum stress amplitude of flexible links; numerical results of the ZOAC model are still close to those of the FOAC model, and the difference between the ZOAC model and FOAC model is very slight.

From the above analysis, we can conclude that the influence of dynamic stiffening of planar 3-RRR flexible parallel robots is very slight and can fully be neglected for elastic dynamic modeling and the conventional ZOAC model can completely reveal dynamic characteristic of the system. The reasons can be explained as follows. (1) Reviewing 𝐊_𝑑=∑_𝑒𝐃^𝑇_𝑒𝐁^𝑇(𝑔_1𝑒𝐇_𝑒+𝐍^𝑇₉𝜑̈−𝐍₈𝜑̇²)𝐁𝐃_𝑒, where 𝜑̈, 𝑔_1𝑒 may take positive or negative value, the values of the first two terms may be positive or negative in bracket, and the third term is positive. Then, the value of the expression in brackets will be small and its sign is indeterminate. After summing with respect to all elements, 𝐊_𝑑 is much smaller than 𝐊₁, so that 𝐊_𝑑 can be neglected in (31). (2) Because of the coupling among flexible links, the influence of dynamic stiffening which is related to rigid-body motion will be weakened.

6. Conclusions

The FOAC model of planar 3-RRR flexible parallel robots with six flexible links was established through the finite element method and the Lagrange equation, in which dynamic stiffening term was considered. The explicit expression of dynamic stiffening was presented. The constraint equations of the system that include the rigid body motion constraints, elastic deformation motion constraints, and dynamics constraints of the moving platform were studied in detail. For comparing the FOAC model with the conventional ZOAC model, numerical simulation was given under two different speeds of the moving platform. The results indicated that the conventional ZOAC model can fully reveal dynamic characteristic of the system and that the influence of dynamic stiffening is very slight and can be neglected for dynamic modeling of planar 3-RRR flexible parallel robots. So, the modeling process can be simplified. Meanwhile, the commercial software ANSYS 13.0 is used to confirm the validity of the zero-order approximation coupling model.

Appendix

𝐩_𝑒,𝑣𝐦̇̇𝜕=−𝐮+1𝜕𝐮2𝐮̇^𝑇𝐦𝐮̇=−𝜑̇²𝐑_𝜑𝜑𝐍₁−𝜑̇²𝐑_𝜑𝜑𝐍₂𝐞_𝑓−2𝜑̇𝐑_𝜑𝐍₂𝐞̇_𝑓−𝜑̇²𝐑_𝜑𝜑𝑉𝜌𝑤_𝑐0𝑑𝑉−𝐑_𝜑𝑉𝜌−𝐞^𝑇_𝑓𝐇_𝑒𝐞̇_𝑓0𝑑𝑉−𝐑_𝑉𝜌−𝐞̇^𝑇_𝑓𝐇_𝑒𝐞̇_𝑓0𝑑𝑉−𝜑̇²𝐑_𝜑𝜑𝐍_𝑐𝐞−2𝜑̇𝐑_𝜑𝐍_𝑐𝐞̇_𝑓−2𝐍₄𝐞̇_𝑓+𝐞^𝑇_𝑓𝐍₈𝐞̇_𝑓+𝐞^𝑇_𝑓𝐍₅𝐞̇_𝑓+𝐞^𝑇𝐍_𝑐𝐞̇_𝑓𝜑̇−𝐞̇^𝑇_𝑓𝐍₇+𝐞̇^𝑇_𝑓𝐍₉+𝐞̇^𝑇_𝑓𝐍_𝑐𝐞̇_𝑓2𝐍₇𝐞̇_𝑓𝜑̇+𝐍^𝑇₄+𝐍₈𝐞_𝑓+𝐍₅𝐞_𝑓𝜑̇²+2𝐍_𝑐𝐞̇_𝑓𝜑̇+𝐍_𝑐𝐞𝜑̇²,𝐌_𝑓(𝜶)=_𝑒𝐃^𝑇_𝑒𝐁^𝑇𝐍₅+𝐂₂𝐁𝐃_𝑒,𝐂_𝑓(𝜶,𝜶̇)=_𝑒𝐃^𝑇_𝑒𝐁^𝑇2𝐍₅+𝐂₂𝐁_𝜑−𝐍₇+𝐍_𝑐𝐁𝜑̇𝐁𝐃_𝑒,𝐐_𝑓(𝜶,𝜶̇,𝜶̈)=_𝑒𝐃^𝑇_𝑒𝐁^𝑇𝐍^𝑇₄𝜑̇²+𝐍_𝑐𝐏₀𝜑̇²−𝐍^𝑇₂+𝐍^𝑇_𝑐𝐑^𝑇𝐫̈_𝑒,𝐴−𝐍^𝑇₆𝜑̈−𝐍^𝑇_𝑐𝐏₀𝜑̈−𝐂^𝑇₁𝜑̈,𝐏₀=[0000𝑙000]^𝑇. (A.1)

Conflict of Interests

The authors declared that they have no conflicts of interest to this work.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant no. 91223201), the Natural Science Foundation of Guangdong Province (Grant no. S2013030013355), Project GDUPS (2010), and the Fundamental Research Funds for the Central Universities (2012ZP0004). This support is greatly acknowledged.