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Research Papers: Applications

Dynamics Model of 4-SPS/CU Parallel Mechanism With Spherical Clearance Joint and Flexible Moving Platform OPEN ACCESS

[+] Author and Article Information
Gengxiang Wang

Faculty of Mechanical and Precision
Instrument Engineering,
Xi'an University of Technology,
P.O. Box 373,
Xi'an, Shaanxi 710048, China
e-mail: wanggengxiang27@163.com

Hongzhao Liu

Faculty of Mechanical and Precision
Instrument Engineering,
Xi'an University of Technology,
P.O. Box 373,
Xi'an, Shaanxi 710048, China
e-mail: liu-hongzhao@163.com

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received December 20, 2016; final manuscript received July 17, 2017; published online September 29, 2017. Assoc. Editor: Sinan Muftu.

J. Tribol 140(2), 021101 (Sep 29, 2017) (10 pages) Paper No: TRIB-16-1395; doi: 10.1115/1.4037463 History: Received December 20, 2016; Revised July 17, 2017

Effects of flexible body and clearance spherical joint on the dynamic performance of 4-SPS/CU parallel mechanism are analyzed. The flexible moving platform is treated as thin plate based on absolute nodal coordinate formulation (ANCF). In order to formulate the parallel mechanism's constraint equations between the flexible body and the rigid body, the tangent frame is introduced to define the joint coordinate. One of the spherical joints between moving platform and kinematic chains is introduced into clearance. The normal and tangential contact forces are calculated based on Flores contact force model and modified Coulomb friction model. The dynamics model of parallel mechanism with clearance spherical joint and flexible moving platform is formulated based on equation of motion. Simulations show that the dynamic performance of parallel mechanism is not sensitive to the flexible body because of the inherent property of moving platform; however, when the clearance spherical joint is considered into the parallel mechanism with flexible body, the flexible moving platform exhibits cushioning effect to absorb the energy caused by clearance joint.

The characteristic of the multibody system in the future will be high-speed, lightweight, precision systems. The parallel mechanism is a special mechanism with the close-loop structure compared to the serial mechanism in the multibody system. In the past three decades, the kinematic and dynamics of parallel mechanism attracted many scholar's attention so that the parallel mechanism can be applied into the various industry environment, such as machining tools [1], space simulator systems [2], food packaging [3], medical application [4], and so on. However, the lightweight components in the parallel mechanism will generate deformation in the case of high operational speed. The accuracy of the parallel mechanism will face a remarkable challenge. Moreover, the clearance existing in the joint is unavoidable because of the manufacturing and assembling tolerances. The clearance joint not only affects the mechanism's kinematics accuracy but also deteriorates the dynamics performance, especially, the contact force caused by clearance joint leads to the vibrations and noise in the motion. If the clearance dimension is very large, the contact force will result in mechanism failure. In general, flexible bodies and clearance joints exist simultaneously.

Regarding the clearance spherical joint in the mechanism, Flores et al. [5] researched the dynamics performance of clearance spherical joint on spatial four-bar mechanism, in order to formulate a precise dynamics model [6]. The lubricated spherical clearance joint is considered in the same spatial mechanism and proved that the presence of a fluid lubricant can alleviate the direct contact between contact bodies. In recent years, they [7] applied the research results of clearance spherical joint into artificial hip joint and obtained that the friction-induced vibration can obviously affect the contact point path during the gait and lead to change in the distribution of contact pressure between ball and socket [8]. Tian et al. [9] studied dynamics of multibody system with flexible link and clearance spherical joint. The flexible link is treated as a spatial beam based on the absolute nodal coordinate formulation (ANCF). Lubrication is introduced into the clearance joint simultaneously. Further, the lubricant and the socket are meshed by finite element based on absolute nodal coordinate formulation. Simulation results are validated by a commercial software [10]. Erkaya et al. [11] built the dynamics model of spatial slider-crank mechanism with clearance spherical joint and proved that the flexible component in this mechanism can absorb the energy caused by clearance joint. Wang et al. [12] predicted the wear depth of clearance spherical join in the four-bar mechanism based on Archard's wear model. However, they mainly focused on kinematic sensitivity [13] and accuracy [14]. Wu and Rao [15] used the interval and fuzzy error analysis method to study the imprecision or uncertainty of mechanism with clearance joints. Frisoli et al. [16] studied the position accuracy of spatial parallel mechanism with revolute clearance joint using the screw theory. Farajtabar et al. [17] thought that the clearance joint releases the kinematic constraint of the ideal joint and proposed a method based on the mass-less virtual model to offset the path error caused by clearance joint. Wang et al. [18] analyzed three-dimensional wear phenomenon of clearance spherical joint in the parallel mechanism based on Archard's wear model.

With respect to the flexible components in the parallel mechanism, Lee and Geng [19] formulated the dynamics model of a flexible steward platform based on the equations of motion. This work provided the fundamental theory for the performance and efficiency of a control algorithm. Kang and Mills [20] built dynamics model of 3-RRR parallel mechanism with flexible link. A piezoelectric actuator located on the surface of the linkages generated a bending moment to damp out the vibration of linkages. Wang and Mills [21] developed the dynamics model of the parallel mechanism with flexible component based on floating frame of reference formulation. Simulation results proved that piezoelectric actuator can effectively damp out the flexible linkage's vibration. Zhang et al. [22] formulated the dynamics model of the 3-PRR (P—prismatic joint and R—revolute joint) parallel mechanism with three flexible links based on the assumed model method; the flexible links are modeled as Euler–Bernoulli beams. In order to verify the correctness of assumed model, experiments were performed using an impact hammer and an accelerometer to identify the mode shapes and frequencies. Test results are consistent with the assumed mode shapes and frequencies. Rezaei et al. [23] used two different methods to research the stiffness property of the 3-PSP (P—prismatic joint and S—spherical joint) parallel mechanism with flexible moving platform. Ebrahimi and Eshaghiyeh-Firoozabadi [24] developed the dynamics performance of three planar flexible manipulators; the flexible links are modeled as the Euler–Bernoulli beams. Sharifnia and Akbarzadeh [25] studied the dynamics performance of a 3-PSP parallel mechanism with flexible moving platform; however, the flexible moving platform is considered as the Euler–Bernoulli beam rather than a plate.

Based on above research, the effects of clearance joints and flexible body on dynamics performance of parallel mechanism are independently studied rather than being considered simultaneously in the same parallel mechanism. The coupling between clearance joint and flexible body is always ignored. Hence, in order to formulate more precise dynamics model, the flexible moving platform and clearance spherical joint are introduced into 4-SPS/CU parallel mechanism at the same time. The moving platform is treated as a thin plate based on ANCF. The spherical joint between moving platform and a certain kinematic chain is taken as clearance joint.

The three-dimensional plate elements are classified as two types in ANCF [26]:

(1) Thick plate element or full parameterized element as shown in Fig. 1(a), the length and width of the quadrilateral plate element are a and b, and it has four nodes (A, B, C, and D). The thickness is t. This element accounts for the transverse shear deformation. The in-plane slope vectors (r/x) and (r/y) are used to define the slope of the midplane at the nodes. The slope vector (r/z) depicts the cross section orientation and deformation. Hence, each node has 12 coordinates; this element is the 48 degrees-of-freedom plate elements:

(2) Thin plate element or gradient deficient element as shown in Fig. 1(b), the length and width of the quadrilateral plate element are a and b. However, since the thickness of this element is small relative to the length and height, this element ignores the thickness and cross section deformation. The element generalized coordinates lack the transverse slope vector (r/z). Therefore, each node has nine coordinates; this element is the 36 degrees-of-freedom plate elements.

Thin Plate Elements.

Based on the structure characteristic of moving platform in parallel mechanism, the thin plate element is adopted to develop the deformable characteristic of moving platform. There are three main reasons: (1) the moving platform is thin and stiff; (2) the elastic force does not include the coupling between the membrane and bending effects so as to eliminate the high frequency [27]; and (3) thin plate element has superior computational efficiency compared to the thick plate element when studying structure with small thickness.

In Fig. 1(b), the element has a total of 36 degrees-of-freedom expressed as follows [28]: Display Formula

(1)e=[r1T,(r1x)T,(r1y)T,r2T,(r2x)T,(r2y)T,r3T,(r3x)T,(r3y)T,r4T,(r4x)T,(r4y)T]

The shape function of the thin plate element can be expressed as follows: Display Formula

(2)S=[s1Is2Is3Is4Is5Is6Is7Is8Is9Is10Is11Is12I]

where I is a 3 × 3 identity matrix, and the shape functions si, (i = 1, 2,…, 12) can be given by Display Formula

(3)s1=(ξ1)(η1)(2η2η+2ξ2ξ1)s2=aξ(ξ1)2(η1),s3=bη(η1)2(ξ1)s4=ξ(2η2η3ξ+2ξ2)(η1)s5=aξ2(ξ1)(η1),s6=bξη(η1)2s7=ξη(13ξ3η+2η2+2ξ2),s8=aξ2η(ξ1)s9=bξη2(η1),s10=η(ξ1)(2ξ2ξ3η+2η2)s11=aξη(ξ1)2,s12=bη2(ξ1)(η1)

where ξ=x/a and η=y/b.

Hence, the global position vector r of an arbitrary point p on the thin plate element in absolute nodal coordinate formulation can be written as Display Formula

(4)r=Se

Equations of Motion of Thin Plate Element.

Based on Kirchhoff theory [29], the strain energy of a thin plate can be expressed as the sum of two terms as shown in Eq. (5). The first term denotes the shear and membrane deformation at the plate midsurface, and the second term represents the energy of the plate bending and twisting Display Formula

(5)U=12VεTEεεdV+12VκTEκκdV

where ε is the Green–Lagrange strain tensor, Eε and Eκ are the elastic coefficient matrix, and κ is the curvature vector [30]. In the case of isotropic plate, the elastic matrix can be written as [29] Display Formula

(6)Eε=Ez1ν2[1ν0ν10002(1ν)]Eκ=z212Eε

in which E is the Young's modulus, v is the Poisson's ratio, and z is the thickness of the thin plate.

The Green–Lagrange strain tensor can be expressed based on the matrix of position vector gradient JDisplay Formula

(7)ε=12(JTJI)

where I is the 3 × 3 identity matrix. Because the thickness of thin plate is ignored, the strain vector at the midsurface can be obtained from Eq. (7), which is expressed as Display Formula

(8)ε=[εxxεyyεxy]T

εxx=(rx)T(rx)12,εyy=(ry)T(ry)12,andεxy=(rx)T(ry)12

where εxx and εyy are the normal strain components in the x and y directions, and εxy is the shear strain.

The curvature vector κ can be written in terms of the gradient vectors [26] as follows: Display Formula

(9)κ=[κxxκyyκxy]T

κxx=(2rx2)Tnn3,κyy=(2ry2)Tnn3,andκxy=(2rxy)Tnn3

where n is the normal vector with respect to the midsurface plate, which can be expressed as n=(r/x)×(r/y).

In absolute nodal coordinate formulation, the elastic force of the thin plate element can be written as Display Formula

(10)Qk=Ue=eTK

where K is the stiffness matrix.

Moreover, in order to obtain the mass matrix of the thin plate element, the kinetic energy of the element can be written as Display Formula

(11)T=12Vρr˙Tr˙dV=12e˙T(VρSTSdV)e˙=12e˙TMee˙

where V is the volume, ρ is the mass density of the thin plate material, and Me is the constant mass matrix of the element.

The equation of motion of the thin plate element can be expressed in matrix form as follows: Display Formula

(12)Mee¨+Qk=Qa

where Qa is the vector of generalized nodal force.

As displayed in Fig. 2, 4-SPS/CU parallel mechanism (S—spherical joint, P—prismatic joint, C—cylindrical joint, and U—universal joint) is organized as four kinematic chains (l1, l2, l3, and l4), a constrained chain l5, moving platform, and base platform. Each kinematic chain consists of a first passive spherical joint, a controlled prismatic joint, and a second passive spherical joint, and a controlled prismatic joint which consists of an actuated rod and a cylinder. The constrained chain l5 consists of a universal joint and a cylindrical joint. Moreover, the two platforms are connected through eight passive spherical joints of four identical kinematic chains. The connection points on the moving platform and base platform are located at A1A2A3A4 and B1B2B3B4, respectively. The side length and the thickness of moving platforms are 2a and h; the side length of based platform is 2b. The structure parameters of the parallel mechanism are presented in Table 1, and the material parameters of the moving platform can be seen in Table 2. The spherical joint located at point A1 is treated as the clearance joint whose parameters are shown in Table 3.

Contact Model of Spherical Joint With Clearance.

As displayed in Fig. 3, in the spherical joint with clearance, A1 is the center of the socket, and b1 is the center of the ball. The radii of ball and socket are Rj and Ri, respectively. The initial radial clearance of spherical joint is defined as c = Ri − Rj. The vectors n and t represent the normal and tangential direction with respect to the contacting surfaces, respectively. Thus, the eccentricity vector d is considered as a vector connecting from point A1 to point b1.

The eccentricity vector d between the socket and the ball in the global coordinate system π0 can be expressed as Display Formula

(13)d=rb1rA1=[dxdydz]T

where the position vectors of points A1 and b1 in the global coordinate system π0 can be expressed as follows: Display Formula

(14){rA1=SA1erb1=r6+R6sb16

where r6 is the position vector of local coordinate system's original point in the global coordinate system, and sb16 is the position vector of ball's center of mass in the local coordinate system π6. R6 is the rotation transformation matrices. SA1 is the shape function, and e is the nodal coordinates of the flexible moving platform.

The magnitude of eccentricity vector d is calculated as Display Formula

(15)d=dTd=dx2+dy2+dz2

A unit vector n vertical to the contacting surfaces between the socket and the ball is expressed as follows: Display Formula

(16)n=dd

When the socket and the ball are in contact, P and Q are the contact points. The position vector in the global coordinate system can be expressed as Display Formula

(17){rP=rA1+nRirQ=rb1+nRj

The contact velocity can be obtained via differentiating Eq. (17) with respect to time, and it is given by Display Formula

(18){r˙P=SA1e˙+n˙Rir˙Q=r˙b1+n˙Rj

The derivative of contact normal vector can be written as Display Formula

(19)n˙=1d[r˙6+R˙6sb16SA1e˙]=1d[r˙6+R6(ω¯6×sb16)SA1e˙]

where r6=[x6y6z6]T is the global coordinate system π0. ω¯6=[ω¯x6ω¯y6ω¯z6]T is angular velocities of body 6 in the local coordinate system π6.

The relative normal and tangential velocities between clearance joint elements can be written as Display Formula

(20){vn=[(r˙Pr˙Q)Tn]nvt=(r˙Pr˙Q)vn

Contact Detection.

The contact deformation can be written as during impact process Display Formula

(21)δ=dc

The contact state can be detected according to Eq. (21); it can be expressed as follows [31]: Display Formula

(22)freeifδ<0startcontactorstartseparateifδ=0contactanddeformationifδ>0

When the magnitude of the eccentricity vector d is less than the radial clearance c, there is noncontact state, and the wear phenomenon between joint elements does not happen. When the magnitude of the eccentricity vector d is equal to the radial clearance c, the contact situation between contact bodies is the critical contact state. When the magnitude of the eccentricity vector e is greater than the radial clearance r, there is contact, the normal and tangential contact forces can be calculated based on the contact force and friction model; however, the wear phenomenon between the joint elements occurs, since the normal contact force is not equal to zero.

Contact Force Model.

The contact force model is divided into two types based on the pure elastic contact force model and dissipative contact force model [32]. The dissipative contact force model is close to the real contact event in the mechanism with clearance joint compared to the pure elastic contact force model. However, the dissipative contact force model, in general, has eight different forms [32]. The coefficient of the restitution and hysteresis damping factor jointly determine the dissipative contact force model. Flores contact force model not only has a relatively simple mathematical structure but also has stable numerical characteristics [33] and has excellent response when the coefficient of the restitution is greater than 0.6. It can be expressed as Display Formula

(23)FN=Kδn[1+8(1cr)5crδ˙δ˙()]

where FN is the contact force, δ is the contact deformation, cr is the coefficient of restitution, δ˙ is the contact deformation velocity, the power exponent n is equal to 3/2 in general when the contact pressure between contact surfaces is considered as parabolic distribution, δ˙() is the initial impact velocity associated with the material properties of contact bodies, in general, δ˙()105E/ρ, ρ is density for contact bodies, and K is a constant depending on the material and geometry properties of contact bodies and it can be expressed as Display Formula

(24)K=43π(σi+σj)(RiRjRiRj)12,σ(k=i,j)=1υk2πEk

where υk is Poisson's ratio of the contact bodies, and Ek is the Young's modulus of the contact bodies.

In order to calculate the tangential contact force between contact bodies, the modified Coulomb friction model [34] is used to compute the friction force, which can be written as Display Formula

(25)Ft=μdcdFNvtvt

where μd is the friction coefficient, and dynamic correction coefficient cd is expressed as Display Formula

(26)cd={0ifvtv0vtv0vmv0ifv0vtvm1ifvtvm

where v0 and vm are considered as tolerances.

In order to formulate the dynamics model of the 4-SPS/CU parallel mechanism with flexible moving platform based on equation of motion, the constraint equations of the parallel mechanism should be formulated first. The parallel mechanism has eight spherical joints, four translation joints, one universal joint, and one cylindrical joint. Because the moving platform is treated as a flexible thin plate element; hence, formulating the parallel mechanism's constraint equation can be classified into two different situations: (i) the constraint equations between rigid and (ii) constraint equations between flexible body and rigid body. The first situation can be referred to the literature [18]. However, the second situation must depend on the joint coordinate system which the orthogonal property is not affected by the deformation of flexible body. Hence, the tangent frame [35] is introduced to define the joint coordinate system.

Tangent Frame.

Formulating the constraint equation corresponding to revolute needs an orthogonal coordinate system to define the constant unit vector. However, in absolute nodal coordinate formulation, the two independent slope coordinate vectors (riek/xie) and (riek/yie) are taken as the nodal coordinates at node k on an element e of body i, which do not remain orthogonal after deformation. Hence, these two slope coordinate vectors cannot be used to define orthogonal axes of the joint coordinate system after deformation. In order to define an orthogonal coordinate system, the tangent frame is introduced into the flexible moving platform [35]. Since the slope vector (riek/xie) remains tangent to the midsurface of the thin plate before and after deformation, the tangent vector regarding the plate midsurface can be defined as Display Formula

(27)tie=rxie|rxie|=r̂xie

where rxie=(rie/xie) and |rxie|=rxieTrxie. A unit vector bie normal to both tie and (rie/yie) can be defined as Display Formula

(28)bie=tie×ryie|ryie|=tie×r̂yie

where ryie=(rie/yie). The other unit vector nie can be expressed as Display Formula

(29)nie=bie×tie

Hence, above unit vector tie, nie, and bie can be used to define the orthogonal transformation matrix, which can be written as Display Formula

(30)Aie=[tieniebie]

In order to satisfy the kinematic constrains at the velocity and acceleration levels, the first and second derivatives of the constraint equation with respect to time should be calculated, which can be seen in the literature [35].

Dynamics Model and Constraint Equations Between Flexible Body and Rigid Body.

The spherical joint is displayed in Fig. 4, the vector Ri is the position vector of center of mass of body li (i = 1,…, 4) in the global coordinate system π0, and the vector u¯Aii is constant vector from the original of body li coordinate system πi to the joint located at point Ai. The vector rAiek denotes the position vector from the global coordinate system π0 to the node k on an element e of moving platform. qi,e are the general coordinates of rigid body and flexible body in the global coordinate system π0, respectively,

Display Formula

(31)C(qi,e)=Ri+Aiu¯AiirAiek=0

The universal joint is displayed in Fig. 5, the vector R5 is the position vector of center of mass of body l5 in the global coordinate system π0, and the vector u¯A55 is constant vector from the original of local coordinate system π5 to the joint located at point A5. The vector rA5ek denotes the position vector from the global coordinate system π0 to the node k on an element e of moving platform. The joint located at node k is attached a joint coordinate system that consists of three orthogonal vector tke, nke, and bke, which can be defined the orthogonal transformation matrix Aie=[tieniebie]. The constraint equations of universal joint between constrained chain l5 and flexible moving platform can be expressed as

Display Formula

(32)C(q5,e)=[R5+A5u¯A55rA5eknekTV15]=0

where V15 is the vector in the global coordinate system, which can be written as V15=A5V¯15. V¯15=[1,0,0]T is the unit vector defined in the local coordinate system π5. nek is the vector in the global coordinate system, which can be written as nek=Aien¯ek. n¯ek=[0,1,0]T is the unit vector defined in the joint coordinate system.

Hence, the constraint equations of the 4-SPS/CU parallel mechanism with clearance spherical joint and flexible moving platform can be written as Display Formula

(33)C=[C(q0,q1),C(q0,q2),C(q0,q3),C(q0,q4)C(q6,q1),C(q7,q2),C(q8,q3),C(q9,q4)C(e,q7),C(e,q8),C(e,q9),C(e,q5),C(q0,q5)]=0

Once the constraint equations are formulated, the dynamics model of parallel mechanism with flexible moving platform can be written as Display Formula

(34)[Mq0CqT0MeCqeTCqrCqe0][q¨e¨λ]=[Qe+QvQaQkQd]

where Mq is the mass matrix of rigid bodies, Me is the mass matrix of flexible body, Cq is the Jacobian matrix of the kinematic constraint equations, q¨ is the acceleration vector of rigid bodies, e¨ is the absolute acceleration vector of flexible body, λ is the Lagrange multipliers vector, and Qe is the generalized external forces that include the system's gravity and contact forces. The contact force is converted into generalized external force, which can be found in the literature [12]. Qv is quadratic velocity that contains gyroscopic moment from differentiating the kinetic energy with respect to time and with respect to the generalized coordinates, Qk is the elastic force of the finite element, Qa is the generalized external nodal forces and contact force, and Qd is the quadratic velocity, which can be obtained by this formulation Cqq¨=Qd.

The expected trajectory of the moving platform is assumed as Fig. 6, which describes revolute motion around y0 axis of the moving platform. Since the contact event occurred in parallel mechanism includes the fast (acceleration) and slow (position) component, in general, this is a stiff problem. Hence, the Runge–Kutta–Fehlberg method is selected as the integrator in this paper. The simulation parameters can be seen in Table 4.

Figures 7 and 8 describe the displacement and velocity of the center of mass of moving platform along the z0 axis. It is obvious that the trajectory is not affected by the clearance spherical joint because the clearance dimension is very small relative to the parallel mechanism dimension. Moreover, the 4-SPS/CU parallel mechanism is not overconstrained so that it has an intrinsic property to offset the manufacturing error to some degree. The velocity is slightly affected by clearance joint because the contact happened in a short time. When the moving platform is considered as flexible body, the displacement and velocity property almost have a same change trend relative to the ideal parallel mechanism, even the acceleration in Fig. 9(a) and reaction force in Fig. 10(a) are consistent with the ideal situation. That is mainly because, although the moving platform is flexible, the material is very stiff (Young's modulus is 71.7 GPa). Hence, the flexible body almost has no influence on the dynamic performance of parallel mechanism [27]. However, in Figs. 9(b) and 10(b), the effect of clearance spherical joint on the acceleration level and reaction force is very significant compared to the position and velocity level. The impact event which occurs in a short time can directly change the magnitude and frequency of the acceleration level and contact force.

From Figs. 1114, there are four scenarios that include the ideal parallel mechanism (ideal model), parallel mechanism with flexible body (flexible model), parallel mechanism with clearance joint (clearance model), and mechanism with clearance joint and flexible body (mixed model). The comparative analysis is implemented in order to explain the effect of flexible moving platform on the dynamic performance of parallel mechanism with clearance spherical joint. In Fig. 11, the displacement and velocity of all scenarios are not sensitive to the clearance spherical joint and flexible moving platform because of the clearance dimension and material property of flexible moving platform. This conclusion is consistent with Figs. 7 and 8. Hence, even if the clearance spherical joint and flexible body are considered in the parallel mechanism simultaneously, the displacement and velocity maintain the same changed trend relative to the ideal parallel mechanism. However, in Figs. 13 and 14, the flexible moving platform has significant effective to damp out high frequency of the contact force and acceleration and reduce its amplitude. That is mainly because the flexible moving platform behaves as a spring–damper element to absorb the energy caused by clearance spherical joint. This behavior will alleviate the vibration and noise of the parallel mechanism in the motion. However, the clearance spherical joint still leads to the high-frequency vibration and increases the reaction force caused by contact in mixed model. The difference between clearance model and mixed model is that the mixed model's magnitude and frequency are smaller than clearance model. That is mainly because of the cushioning function of flexible moving platform. The phenomenon accounts for the coupling relationship between flexible body and clearance spherical joint in the parallel mechanism, because the contact force in dynamics model Eq. (34) includes the elastic force from the flexible moving platform. The same situation can be seen in literature [9,11,24].

Since the dynamics model of the flexible moving platform is formulated based on the thin plate element using absolute nodal coordinate formulation, the stress of center of mass of the moving platform just includes normal stress along x0 and y0 axis and shear stress. That is because that the thin plate ignores the thickness and cross section deformation so that the stresses along z0 axis do not exist. From Figs. 15 and 16, although the stress also displayed the high-frequency property caused by contact, the magnitude of the stress of the mixed model is close to the stress of the Flexible model, because the flexible moving platform has much greater Young's modulus, which lead to smaller deformation so that the magnitude of stress has not increased significantly. This phenomenon proves that the vibration of the clearance model and mixed model is from the contact between socket and ball, which account for the correctness of dynamics model of 4-SPS/CU parallel mechanism with flexible moving platform and clearance spherical joint.

The objective of this paper is to develop the effect of the flexible moving platform and clearance spherical joint on dynamic performance of the 4-SPS/CU parallel mechanism. The flexible moving platform is treated as the thin plate element based on absolute nodal coordinate formulation. One of the spherical joints is introduced into clearance. The kinematics model of clearance spherical joint is described to obtain the contact deformation and contact velocity. The normal contact force and the tangential force are calculated based on Flores contact force model and modified Coulomb friction model. In order to build constraint equation between the flexible moving platform and the kinematic chain, the tangent frame is used to define the joint coordinate system where the orthogonality is not affected by deformation. The dynamics model of the parallel mechanism with flexible moving platform and clearance spherical joint was built based on the equation of motion with constraint equations.

Simulation results using the flexible mode show that the dynamic performance of the parallel mechanism is not significantly affected by flexible moving platform, since its material property results in the small deformation. Regarding the clearance model, the position level is not sensitive with the clearance spherical joint because of the clearance dimension and structure characteristics of 4-SPS/CU parallel mechanism. The velocity level has a slight change, since contact happens in a short time. In contrast, the acceleration level and the reaction force display high frequency with sudden magnitude change caused by the clearance spherical joint. Regarding the mixed model, the flexible moving platform presents a suspension effect to absorb the energy from the contact between socket and ball, which can filter out the high frequency and reduce the magnitude of acceleration level and reaction force. Hence, the flexible property of moving platform cannot be neglected in the parallel mechanism with clearance spherical joint. Likewise, the effect of clearance spherical joint on flexible model should be noted; both shear stress and normal stress are affected by contact between socket and ball.

  • The State Scholarship Fund organized by the National Natural Science Foundation of China (Grant No. 51275404).

  • The China Scholarship Council (CSC) in 2015 (Grant No. 201508610101).

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Erkaya, S. , Doğan, S. , and Şefkatlıoğlu, E. , 2016, “ Analysis of the Joint Clearance Effects on a Compliant Spatial Mechanism,” Mech. Mach. Theory, 104, pp. 255–273.
Wang, G. , Liu, H. , and Deng, P. , 2015, “ Dynamics Analysis of Spatial Multibody System With Spherical Joint Wear,” ASME J. Tribol., 137(2), p. 021605.
Tannous, M. , Caro, S. , and Goldsztejn, A. , 2014, “ Sensitivity Analysis of Parallel Manipulators Using an Interval Linearization Method,” Mech. Mach. Theory, 71, pp. 93–114.
Chaker, A. , Mlika, A. , Laribi, M. A. , Romdhane, L. , and Zeghloul, S. , 2013, “ Clearance and Manufacturing Errors' Effects on the Accuracy of the 3-RCC Spherical Parallel Manipulator,” Eur. J. Mech. A/Solids, 37, pp. 86–95.
Wu, W. , and Rao, S. S. , 2004, “ Interval Approach for the Modeling of Tolerances and Clearances in Mechanism Analysis,” ASME J. Mech. Des., 126(4), pp. 581–592.
Frisoli, A. , Solazzi, M. , Pellegrinetti, D. , and Bergamasco, M. , 2011, “ A New Screw Theory Method for the Estimation of Position Accuracy in Spatial Parallel Manipulators With Revolute Joint Clearances,” Mech. Mach. Theory, 46(12), pp. 1929–1949.
Farajtabar, M. , Daniali, H. M. , and Varedi, S. M. , 2016, “ Error Compensation for Continuous Path Operation of 3-RRR Planar Parallel Manipulator With Clearance in the Joints,” J. Sci. Eng., 7(1), pp. 20–33.
Wang, G. , Liu, H. , Deng, P. , Yin, K. , and Zhang, G. , 2017, “ Dynamic Analysis of 4-SPS/CU Parallel Mechanism Considering Three-Dimensional Wear of Spherical Joint With Clearance,” ASME J. Tribol., 139(2), p. 021608.
Lee, J. D. , and Geng, Z. , 1993, “ A Dynamic Model of a Flexible Stewart Platform,” Comput. Struct., 48(3), pp. 367–374.
Kang, B. , and Mills, J. K. , 2002, “ Dynamic Modeling of Structurally-Flexible Planar Parallel Manipulator,” Robotica, 20(3), pp. 329–339.
Wang, X. , and Mills, J. K. , 2005, “ FEM Dynamic Model for Active Vibration Control of Flexible Linkages and Its Application to a Planar Parallel Manipulator,” Appl. Acoust., 66(10), pp. 1151–1161.
Zhang, X. , Mills, J. K. , and Cleghorn, W. L. , 2007, “ Dynamic Modeling and Experimental Validation of a 3-PRR Parallel Manipulator With Flexible Intermediate Links,” J. Intell. Robot. Syst: Theory Appl., 50(4), pp. 323–340.
Rezaei, A. , Akbarzadeh, A. , and Akbarzadeh-T, M. R. , 2012, “ An Investigation on Stiffness of a 3-PSP Spatial Parallel Mechanism With Flexible Moving Platform Using Invariant Form,” Mech. Mach. Theory, 51, pp. 195–216.
Ebrahimi, S. , and Eshaghiyeh-Firoozabadi, A. , 2016, “ Dynamic Performance Evaluation of Serial and Parallel RPR Manipulators With Flexible Intermediate Links,” Iran. J. Sci. Technol. Trans. Mech. Eng., 40(3), pp. 169–180.
Sharifnia, M. , and Akbarzadeh, A. , 2016, “ Approximate Analytical Solution for Vibration of a 3-PRP Planar Parallel Robot With Flexible Moving Platform,” Robotica, 34(1), pp. 71–97.
Hyldahl, P. , Mikkola, A. M. , Balling, O. , and Sopanen, J. T. , 2014, “ Behavior of Thin Rectangular ANCF Shell Elements in Various Mesh Configurations,” Nonlinear Dyn., 78(2), pp. 1277–1291.
Shabana, A. A. , 2011, Computational Continuum Mechanics, Cambridge University Press, New York.
Dufva, K. , and Shabana, A. A. , 2005, “ Analysis of Thin Plate Structures Using the Absolute Nodal Coordinate Formulation,” Proc. Inst. Mech. Eng., Part K, 219(4), pp. 345–355.
Bathe, K. J. , 2006, Finite Element Procedures, Prentice Hall, Upper Saddle River, NJ.
Dmitrochenko, O. N. , and Pogorelov, D. Y. , 2003, “ Generalization of Plate Finite Elements for Absolute Nodal Coordinate Formulation,” Multibody Syst. Dyn., 10(1), pp. 17–43.
Flores, P. , and Ambrósio, J. , 2010, “ On the Contact Detection for Contact-Impact Analysis in Multibody Systems,” Multibody Syst. Dyn., 24(1), pp. 103–122.
Machado, M. , Moreira, P. , Flores, P. , and Lankarani, H. M. , 2012, “ Compliant Contact Force Models in Multibody Dynamics: Evolution of the Hertz Contact Theory,” Mech. Mach. Theory, 53, pp. 99–121.
Flores, P. , Machado, M. , Silva, M. T. , and Martins, J. M. , 2011, “ On the Continuous Contact Force Models for Soft Materials in Multibody Dynamics,” Multibody Syst. Dyn., 25(3), pp. 357–375.
Ambrósio, J. A. C. , 2003, “ Impact of Rigid and Flexible Multibody Systems: Deformation Description and Contact Models,” Virtual Nonlinear Multibody Systems, Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 57–81.
Sugiyama, H. , Escalona, J. L. , and Shabana, A. A. , 2003, “ Formulation of Three-Dimensional Joint Constraints Using the Absolute Nodal Coordinates,” Nonlinear Dyn., 31(2), pp. 167–195.
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References

Tian, Y. , Shirinzadeh, B. , and Zhang, D. , 2010, “ Design and Dynamics of a 3-DOF Flexure-Based Parallel Mechanism for Micro/Nano Manipulation,” Microelectron. Eng., 87(2), pp. 230–241.
Yang, C. , Huang, Q. , Jiang, H. , Peter, O. O. , and Han, J. , 2010, “ PD Control With Gravity Compensation for Hydraulic 6-DOF Parallel Manipulator,” Mech. Mach. Theory, 45(4), pp. 666–677.
Zhang, J. Z. , Xie, X. P. , Li, C. J. , Xin, Y. Y. , and He, Z. M. , 2013, “ FEM Based Numerical Simulation of Delta Parallel Mechanism,” Adv. Mater. Res., 690–693, pp. 2978–2981.
Ottoboni, A. , Parenti-Castelli, V. , Sancisi, N. , Belvedere, C. , and Leardini, A. , 2010, “ Articular Surface Approximation in Equivalent Spatial Parallel Mechanism Models of the Human Knee Joint: An Experiment-Based Assessment,” Proc. Inst. Mech. Eng., Part H, 224(9), pp. 1121–1132.
Flores, P. , Ambrósio, J. , Claro, J. C. , and Lankarani, H. M. , 2006, “ Dynamics of Multibody Systems With Spherical Clearance Joints,” ASME J. Comput. Nonlinear Dyn., 1(3), pp. 240–247.
Flores, P. , and Lankarani, H. M. , 2010, “ Spatial Rigid-Multibody Systems With Lubricated Spherical Clearance Joints: Modeling and Simulation,” Nonlinear Dyn., 60(1–2), pp. 99–114.
Askari, E. , Flores, P. , Dabirrahmani, D. , and Appleyard, R. , 2014, “ Study of the Friction-Induced Vibration and Contact Mechanics of Artificial Hip Joints,” Tribol. Int., 70, pp. 1–10.
Askari, E. , Flores, P. , Dabirrahmani, D. , and Appleyard, R. , 2014, “ Nonlinear Vibration and Dynamics of Ceramic on Ceramic Artificial Hip Joints: A Spatial Multibody Modelling,” Nonlinear Dyn., 76(2), pp. 1365–1377.
Tian, Q. , Zhang, Y. , Chen, L. , and Flores, P. , 2009, “ Dynamics of Spatial Flexible Multibody Systems With Clearance and Lubricated Spherical Joints,” Comput. Struct., 87(13–14), pp. 913–929.
Tian, Q. , Lou, J. , and Mikkola, A. , 2017, “ A New Elastohydrodynamic Lubricated Spherical Joint Model for Rigid-Flexible Multibody Dynamics,” Mech. Mach. Theory, 107, pp. 210–228.
Erkaya, S. , Doğan, S. , and Şefkatlıoğlu, E. , 2016, “ Analysis of the Joint Clearance Effects on a Compliant Spatial Mechanism,” Mech. Mach. Theory, 104, pp. 255–273.
Wang, G. , Liu, H. , and Deng, P. , 2015, “ Dynamics Analysis of Spatial Multibody System With Spherical Joint Wear,” ASME J. Tribol., 137(2), p. 021605.
Tannous, M. , Caro, S. , and Goldsztejn, A. , 2014, “ Sensitivity Analysis of Parallel Manipulators Using an Interval Linearization Method,” Mech. Mach. Theory, 71, pp. 93–114.
Chaker, A. , Mlika, A. , Laribi, M. A. , Romdhane, L. , and Zeghloul, S. , 2013, “ Clearance and Manufacturing Errors' Effects on the Accuracy of the 3-RCC Spherical Parallel Manipulator,” Eur. J. Mech. A/Solids, 37, pp. 86–95.
Wu, W. , and Rao, S. S. , 2004, “ Interval Approach for the Modeling of Tolerances and Clearances in Mechanism Analysis,” ASME J. Mech. Des., 126(4), pp. 581–592.
Frisoli, A. , Solazzi, M. , Pellegrinetti, D. , and Bergamasco, M. , 2011, “ A New Screw Theory Method for the Estimation of Position Accuracy in Spatial Parallel Manipulators With Revolute Joint Clearances,” Mech. Mach. Theory, 46(12), pp. 1929–1949.
Farajtabar, M. , Daniali, H. M. , and Varedi, S. M. , 2016, “ Error Compensation for Continuous Path Operation of 3-RRR Planar Parallel Manipulator With Clearance in the Joints,” J. Sci. Eng., 7(1), pp. 20–33.
Wang, G. , Liu, H. , Deng, P. , Yin, K. , and Zhang, G. , 2017, “ Dynamic Analysis of 4-SPS/CU Parallel Mechanism Considering Three-Dimensional Wear of Spherical Joint With Clearance,” ASME J. Tribol., 139(2), p. 021608.
Lee, J. D. , and Geng, Z. , 1993, “ A Dynamic Model of a Flexible Stewart Platform,” Comput. Struct., 48(3), pp. 367–374.
Kang, B. , and Mills, J. K. , 2002, “ Dynamic Modeling of Structurally-Flexible Planar Parallel Manipulator,” Robotica, 20(3), pp. 329–339.
Wang, X. , and Mills, J. K. , 2005, “ FEM Dynamic Model for Active Vibration Control of Flexible Linkages and Its Application to a Planar Parallel Manipulator,” Appl. Acoust., 66(10), pp. 1151–1161.
Zhang, X. , Mills, J. K. , and Cleghorn, W. L. , 2007, “ Dynamic Modeling and Experimental Validation of a 3-PRR Parallel Manipulator With Flexible Intermediate Links,” J. Intell. Robot. Syst: Theory Appl., 50(4), pp. 323–340.
Rezaei, A. , Akbarzadeh, A. , and Akbarzadeh-T, M. R. , 2012, “ An Investigation on Stiffness of a 3-PSP Spatial Parallel Mechanism With Flexible Moving Platform Using Invariant Form,” Mech. Mach. Theory, 51, pp. 195–216.
Ebrahimi, S. , and Eshaghiyeh-Firoozabadi, A. , 2016, “ Dynamic Performance Evaluation of Serial and Parallel RPR Manipulators With Flexible Intermediate Links,” Iran. J. Sci. Technol. Trans. Mech. Eng., 40(3), pp. 169–180.
Sharifnia, M. , and Akbarzadeh, A. , 2016, “ Approximate Analytical Solution for Vibration of a 3-PRP Planar Parallel Robot With Flexible Moving Platform,” Robotica, 34(1), pp. 71–97.
Hyldahl, P. , Mikkola, A. M. , Balling, O. , and Sopanen, J. T. , 2014, “ Behavior of Thin Rectangular ANCF Shell Elements in Various Mesh Configurations,” Nonlinear Dyn., 78(2), pp. 1277–1291.
Shabana, A. A. , 2011, Computational Continuum Mechanics, Cambridge University Press, New York.
Dufva, K. , and Shabana, A. A. , 2005, “ Analysis of Thin Plate Structures Using the Absolute Nodal Coordinate Formulation,” Proc. Inst. Mech. Eng., Part K, 219(4), pp. 345–355.
Bathe, K. J. , 2006, Finite Element Procedures, Prentice Hall, Upper Saddle River, NJ.
Dmitrochenko, O. N. , and Pogorelov, D. Y. , 2003, “ Generalization of Plate Finite Elements for Absolute Nodal Coordinate Formulation,” Multibody Syst. Dyn., 10(1), pp. 17–43.
Flores, P. , and Ambrósio, J. , 2010, “ On the Contact Detection for Contact-Impact Analysis in Multibody Systems,” Multibody Syst. Dyn., 24(1), pp. 103–122.
Machado, M. , Moreira, P. , Flores, P. , and Lankarani, H. M. , 2012, “ Compliant Contact Force Models in Multibody Dynamics: Evolution of the Hertz Contact Theory,” Mech. Mach. Theory, 53, pp. 99–121.
Flores, P. , Machado, M. , Silva, M. T. , and Martins, J. M. , 2011, “ On the Continuous Contact Force Models for Soft Materials in Multibody Dynamics,” Multibody Syst. Dyn., 25(3), pp. 357–375.
Ambrósio, J. A. C. , 2003, “ Impact of Rigid and Flexible Multibody Systems: Deformation Description and Contact Models,” Virtual Nonlinear Multibody Systems, Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 57–81.
Sugiyama, H. , Escalona, J. L. , and Shabana, A. A. , 2003, “ Formulation of Three-Dimensional Joint Constraints Using the Absolute Nodal Coordinates,” Nonlinear Dyn., 31(2), pp. 167–195.

Figures

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Fig. 1

Four nodes plate element

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Fig. 2

4-SPS/CU parallel mechanism

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Fig. 3

Contact kinematic of spherical joint with clearance

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Fig. 4

Spherical joint between rigid body and flexible body

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Fig. 5

Universal joint between rigid body and flexible body

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Fig. 6

Trajectory of the moving platform

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Fig. 7

Effect of the flexible component and clearance spherical joint on displacement

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Fig. 8

Effect of the flexible component and clearance spherical joint on velocity

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Fig. 9

Effect of the flexible component and clearance spherical joint on acceleration

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Fig. 10

Effect of the flexible component and clearance on reaction force of spherical joint

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Fig. 11

Displacement comparison

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Fig. 12

Velocity comparison

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Fig. 13

Acceleration comparison

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Fig. 14

Contact force comparison

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Fig. 15

Shear stress comparison

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Fig. 16

Normal stress comparison

Tables

Table Grahic Jump Location
Table 1 Physical parameters of 4-SPS/CU parallel mechanism
Table Grahic Jump Location
Table 2 Material parameter of moving platform
Table Grahic Jump Location
Table 3 Physical parameters of spherical clearance joint
Table Grahic Jump Location
Table 4 Simulation parameters

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