Kinetostatic modelling of a 3-PRR planar compliant parallel manipulator with flexure pivots

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Kinetostatic modelling of a 3-PRR planar compliant parallel manipulator with flexure pivots Miao Yang a , Zhijiang Du a , Fangxin Chen a , Wei Dong a,∗ , Dan Zhang b a b

State Key Laboratory of Robotics and System, Harbin Institute of Technology, 2 Yikuang Street, Harbin, 150080, China Department of Mechanical Engineering, Lassonde School of Engineering, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada

a r t i c l e

i n f o

Article history: Received 9 October 2016 Received in revised form 26 December 2016 Accepted 3 January 2017 Available online xxx Keywords: Flexure pivot Compliant parallel manipulator Kinetostatic analysis Large deformation

a b s t r a c t Flexure pivots are widely used in long stroke complaint parallel manipulators (CPMs) for their large deformation capacity. However the parasitic motion caused by the flexure pivots cannot be neglected as the rotation angle gets large, which will finally affect the absolute positioning accuracy of the manipulators. In this paper, a modelling approach to calculate the nonlinear kinetostatics of a long stroke planar CPM with flexure pivots is proposed, in which the parasitic motion of the flexure joints are taken into account. The displacement constraint equations and the static equilibrium equations of the CPM are established under the deformed configuration. Finite element analysis (FEA) shows the accuracy and efficiency of the nonlinear kinematic model (NLKM) for both the inverse and forward kinematic analysis. Moreover, the prediction accuracy of the CPM’s kinematic behavior has been improved more than 20 times by employing the NLKM compared with the conventional kinematic model (CKM). © 2017 Elsevier Inc. All rights reserved.

1. Introduction Planar parallel manipulators (PPMs) have many advantages, e.g., low inertia, compact structure, superior dynamic performance and high precision [1,2]. However, the motion accuracy of PPMs can be further improved by adopting flexure joints to transmit forces and motions instead of using conventional rotation joints, since flexure joints are naturally free of backlash, friction and slip-stick effects [3,4]. Compliant PPMs are widely used in high-precision manufacturing tools, measurement devices and scientific instruments to implement precision motion tasks [5,6]. Conventional notch-type flexure joints suffer from stress concentration, the motion range of CPMs with notched flexure joints is mainly limited in micro/nano scale which restricts the application of CPMs seriously [7–9]. This issue can be solved by constructing CPMs with distributed-compliance flexure joints which feature large deformation capacities [10–12]. The flexure pivot is a typical distributed-compliance flexure joint, formed by a fixed rigid part, a moving rigid part and two identical elastic leaves crossed (not connected) at the midpoint along the leaf length. The spring leaves are the elastic elements of the flexure joint, and external loads can

∗ Corresponding author. E-mail address: [email protected] (W. Dong).

easily make the moving parts deflect with an angle of 20◦ since stresses are evenly distributed on the structure [13–15]. Although the distributed-compliance based CPMs have advantages in motion range, the parasitic motion caused by the flexure joints’ center shift can not be neglected especially when the rotation angle is large [16]. Taking the commercial available flexure pivot manufactured by C-Flex (J-10) for example, the designed rotation center of a flexure pivot is located at the cross point, while the actual rotation center is difficult to maintain its position due to the distributed compliance, the center shift can be as much as 122 ␮m when the pivot rotates by 10◦ [17]. This deviation will be translated into the whole compliant mechanism and eventually affect the absolute positioning accuracy. To improve the open loop motion performance of CPMs, there are always two kinds of methods. The first approach is to apply flexure elements with appropriate arrangements that can reduce the parasite motion [18–20]. Another method is to employ common flexure joints, but to establish accuracy kinetostatic models for CPMs in which the motion error of flexure joints are considered [21–23]. Recent years many researchers proposed a few of creative modified flexure pivot structures to minimize the parasitic motion during the deformation, but those flexure pivot are always with complex structures and hard to eliminate the parasitic motion completely when they suffer from relatively larger lateral forces [15,17,24–26]. Therefore, establishing kinetostatic models for the

http://dx.doi.org/10.1016/j.precisioneng.2017.01.002 0141-6359/© 2017 Elsevier Inc. All rights reserved.

Please cite this article in press as: Yang M, et al. Kinetostatic modelling of a 3-PRR planar compliant parallel manipulator with flexure pivots. Precis Eng (2017), http://dx.doi.org/10.1016/j.precisioneng.2017.01.002

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Fig. 1. 3-PRR planar parallel manipulator.

flexure based CPMs may be a more generalized and effective method to improve the motion accuracy of CPMs. A lot of efforts also have been made in this field, Yuen [27] derived a kinetostatic model for a 3-RRR compliant nano-positioner with right circular flexure hinges by using the compliance transformation matrix method. Su [28] formulated an analytical model for calculating the kinematics of a flexure hexapod nano-positioner based on the screw theory. Rouhani [29] established a kinetostatic model for a 6-DOF compliant micro-hexapod with spatial beam type flexure joints. However, all those methods only work for CPMs under small deformation. Li [30] proposed a constraint-force-based approach to calculate the deformation of a decoupled XYZ compliant parallel mechanism under an intermediate motion range by using the nonlinear closedform spatial beam model proposed by Hao [31]. For modelling the large deformed CPMs, previous studies are mainly focused on finite element method (FEM), e.g., Dong [32] analyzed the stiffness characteristics of a CPM utilizing the nonlinear FEM, and Yun [33] employed FEM to optimize the structure of a 3-PUPU CPM. In this paper, we proposed an accurate modelling approach to calculate the nonlinear kinetostatics of a long stroke 3-PRR planar CPM with flexure pivots, which can be used to improve the absolute positioning accuracy of the CPM. The displacement constraint equations and the static equilibrium equations of the CPM are established by considering the large deformed configuration and the parasite motion of the flexure joints. The remainder of the paper is organized as follows, the structure and the conventional kinematic model of the CPM is firstly described in Section 2. The displacement constraint equations and the static equilibrium equations of the CPM are established under the deformation configuration and presented in Section 3. In Section 4, the proposed method has been verified by FEA and the kinematic analysis of the CPM is also conducted. Finally, the paper is concluded in Section 5.

Fig. 2. 3-PRR compliant parallel manipulator with flexure pivots.

of freedom motion in the xy planar, i.e., two translation and one rotation. The coordinate Oxy is the global coordinate system (GCS) of the manipulator which is fixed at the center of the base platform G1 G2 G3 , and the coordinate Pxy is a local coordinate system (LCS) located at the center of the moving platform which moves and rotates with the moving platform I1 I2 I3 . A straightforward kinematic replacement method is adopted to construct the CPM by using flexure pivots replacing the conventional passive rotation joints in the 3-PRR parallel manipulator. The cross points of the spring leaves are assigned at the original rotation center of the passive rotation joints, i.e., points Hi (i = 1, 2, 3) in Fig. 1, and the axial direction of the flexure pivots are followed with the passive links Hi Ii . As shown in Fig. 2, the flexure pivots P1i are connected with the intermediate links Bi Ci on one end, and the other end is fixed with the moving slider at point Di . Similarly, the flexure pivots P2i are connected to the intermediate links Bi Ci on one end, and bonds with the moving platform at point Ai on the other ends. In the conventional kinematic model, all the flexure joints are considered as ideal rotational joints and the parasitic motions of the flexure joints are ignored. Both the solution of the inverse kinematic model (IKM) and direct kinematic model (DKM) can be easily obtained through the closed loop vector equations of each kinematic chain in GCS [34], as below, OP + PIi = OGi + Gi Hi + Hi Ii

This vector equation can be rewritten algebraically in the Cartesian space, as follows



xP yP





2. Conventional kinematic analysis The architecture of the planar mechanism investigated in this paper is 3-PRR, which consists of three identical limbs, and each limb is comprised of an active prismatic joint (P) and two passive rotation joints (R) as shown in Fig. 1. Both the moving platform I1 I2 I3 and the base platform G1 G2 G3 are equilateral triangle-shaped, and the centers of the two platforms coincide at point O initially with a declination angle ϕp . The moving platform can realize 3 degrees

(1)

=









+ R ϕp + ϕ xHi0 yHi0

+ i



x Ii y Ii

cos ˛i sin ˛i





 + Ldi

cos ˇi

 (2)

sin ˇi

where, xP and yP are the displacement of the moving platform in x-axis and y-axis direction respectively, ␸ is the rotation angle of the moving platform from the initial pose, x’Ii and y’Ii are the coordinates of point Ii in the LCS, xHi0 and yHi0 are the initial coordinates of point Hi in the GCS, ␳i is the input displacement of the actuator, ␣i is the orientation angle of the actuator, ld is the length of the

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Fig. 4. Deformed configuration of the flexure pivot.

Fig. 3. Geometry description of the flexure pivot.

over, L, T and W denote the length, thickness and width of the leaf respectively, and 2␤ is the cross angle of the spring leaves. Fig. 4 shows the deformed configuration of a flexure pivot. The deformation of the flexure pivot only occurs on the spring leaves, and each spring leaves can be abstracted as cantilever beams subjected combined forces at the free ends. In LCS Pixy (i = 1, 2), the spring leaf is subject to a transverse force Fbxi , an axial force Fbyi and a moment Mbzi , and the coordinate of the end point of the leaf is xbi and ybi , ␪bi is the deflection angle of the leaf. The corresponding load-displacement relationships of the spring leaves can be obtained by using Awart’s beam model [36].

intermediate link Hi Gi , ␤i is the rotation angle of the intermediate to the global x-axis, and the transformation links Hi Gi with respect  matrixR ϕp + ϕ can be defined as





R ϕp + ϕ =



 

 

cos ϕp + ϕ



− sin ϕp + ϕ







sin ϕp + ϕ cos ϕp + ϕ For the inverse kinematics, the displacement of the actuator ␳i should be found for a given pose of the moving platform (xP , yP ,ϕ). And the forward kinematics is the opposite case, i.e., given



fi mi

  =

12

−6

−6

4

T 2 pi 1 ıxi = − ı 2 yi 12L2



ıyi bi

bi





3



 +pi

−1/10

6/5



−1/10 2/15

−3/5

1/20

1/20

−1/15



the displacement of the actuators to find out the pose of the moving platform. Although Eq. (2) gives a quick kinematic modelling of the parallel manipulator, the prediction errors of the CKM will be very large and affect the absolute positioning accuracy of the CPM for large deformation (as shown in Section 4). In order to overcome this problem, the parasitic motion of the flexure joints must be discussed in detail.

ıyi

3.1. Load–displacement relationship of the flexure pivot To formulate a kinetostatic model for the 3-PRR CPM, the first step is to describe the load-displacement relationship of the flexure pivot, a deflection model of the flexure pivot similar to that proposed in [13,35] is employed here for completeness. Fig. 3 illus-

bi





− pi ıyi

bi

bi





−1/1400

1/700



−1/1400 11/6300

ıyi



Fbyi L2

fi =

EI

Fbxi L2 EI

pi =

Mbzi L L − xbi y ıxi = ıyi = bi EI L L

Moreover, the derivation of stresses on the spring leaves can be found in [35]. For the sake of simplicity, it is not included here. Based on the positional relation depicted in Fig. 4, the closed loop equations are given as,

   x  b2 0 =

R ˇ

yb2 t = b1 , t = b2



 

+ R −ˇ

L sin ˇ



xb1





+ L sin ˇ

yb1

− sin t cos t



(4)

where t is the rotation angle of the flexure pivot. The static equilibrium equations of a pivot is defined as below.

⎧      P −Fbx1 −Fbx2 ⎪     ⎪ ⎪ ⎪ + R −ˇ +R ˇ =0 ⎪ ⎨ V −Fby1 −Fby2         − sin t P − sin t ⎪   −Fbx2 ⎪ L ⎪ ⎪ M − Mbz1 − Mbz2 + sin ˇ × + L sin ˇ ×R ˇ =0 ⎪ ⎩ 2 V

cos t

trates the geometry description of the flexure pivot, where the global coordinate system P0 xy is located at P0 and x-axis follows along the axial direction of the pivot, P1 xl yl and P2 xl yl are two fixed local coordinates with x-axis along the leaf length direction. More-

(3)

bi

In the above equation, fi , pi and mi are the nondimensional transverse and axial forces and the end moments applied on the spring leaf, and ␦x and ␦y are the nondimensional transverse and axial displacements at the end of the leaf, which are defined as below,

mi = 3. Nonlinear kinematic modelling



ıyi

(5)

−Fby2

cos t

The coordinates at the center of the moving part is given by the formulation as below.



xt yt



1   = R −ˇ 2



xb1 yb1



1   + R ˇ 2



xb2



(6)

yb2

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respect to the x-axis of GCS is a constant value and can be calculated in advance because the pivot and the slider are rigidly connected. Bi xy is a moving frame to describe the deflection of pivot P2i , its origin is chosen at point Bi , and the y-axis follows the transverse direction of pivot P2i .  D D In LCS Di xy the end point of the limb Ai xAi , yAi can be formulated as below,



D xAi





=

D yAi



xt1i



+ lbc

yt1i

cos t1i





+ R t1i

sin t1i





xt2i



(10)

yt2i

where xt1i , yt1i are the displacement of pivot P1i in frame Di xy, t1i is the rotation angle of the pivot, xt2i , yt2i are the displacement of pivot P2i in frame Bi xy, and lbc is the length of the link Bi Ci which can be calculated as below. lbc = ld − 2L cos ˇ

(11)

Then the coordinates of Ci (xCi , yCi ) and Ai (xAi , yAi ) on the limb in GCS can be expressed as follows.

Fig. 5. Deformed configuration of the 3-PRR CPM.



xCi



=

yCi



xAi









xDi yDi



+ R (i )

yDi

=

yAi

xDi





xt1i

(12)

yt1i

 + R (i )

Dx

 Ai

Dy

(13)

Ai

Obviously, the coordinates of Ai represented by Eqs. (8) and (13) should be the same, thus we can obtain the kinematic equation of the limb.

⎧ m x −x =0 ⎪ ⎨ Ai Ai ⎪ ⎩

Fig. 6. Deformed configuration of the kinematic limb.

Combining Eqs. (3)–(6), the load-displacement relationship of the flexure pivot is obtained. Since it is hard to be expressed explicitly, a general function  is used to represent the mapping  from the external loads (P, V, M) to the deformation of the pivot xt , yt , t as below.





xt , yt , t =  (P, V, M)

(7)

3.2. Formulation of nonlinear kinematic equations



m xAi





=

m yAi

xp



yp



+ R (ϕ)

xAi0



(8)



xDi yDi





=

xDi0 yDi0





+ i

cos ˛i



(9)

sin ˛i

where xDi0 , and yDi0 denote the initial coordinates of the point Di in the undeformed configuration. To present the deflection of the flexure pivots, two kinds of local coordinate systems should be introduced, where Di xy is a moving frame attached on the point Di , and the y-axis coincides with the rigid part of the pivot P1i . The deflection angle ␥i of the pivot with

ϕ − t1i − t2i = 0

thus



the

intermediate

forces

T

F1i =



Fx1i , Fy1i , Mz1i

T

and

F2i = Fx2i , Fy2i , Mz2i applied on pivot P1i and pivot P2i in GCS are introduced. According to the static equilibrium equations of the limb the components of F2i is given as below. Fx2i

⎞

⎛

1

⎜ ⎟ ⎝ Fy2i ⎠ = ⎝ Mz2i

⎞⎛F ⎞ x1i ⎟ ⎜ ⎠ 0 ⎝ Fy1i ⎠

0

0

0

1

− (yAi − yCi )

(xAi − xCi )

1

(15)

Mz1i

Based on the load-displacement relationship developed above, the deflection for flexure pivot P1i can be formulated as,



yAi0

where xAi0 and yAi0 are the initial coordinates of the point Ai in GCS. The coordinates of the point Di (xDi ,yDi ) in the GLS is expressed by

(14)

Eq. (14) is a system of nonlinear equations and it the  deflection of the flexure cannot be  solved unless T T pivots xt1i , yt1i , t1i and xt2i , yt2i , t2i are known,

⎛

The deformed configuration of the 3-PRR CPM is presented in Fig. 5. In order to facilitate the description, a single kinematic limb of the compliant 3-PRR mechanism is  taken into account as shown in m , ym on the moving platform Fig. 6. The coordinates of point Ai xAi Ai in the GCS can be given as below,

m −y =0 yAi Ai





xt1i , yt1i , t1i =  Pp1i , Vp1i , Mp1i



(16)

wherePp1i is the axial force applied to the pivot in LCS Di xy, Vp1i is the vertical force applied at the end of the pivot and the Mp1i is the moment, which can be obtained through the following transformation.

⎛

Pp1i

⎞

⎛

cos i

sin i

0

⎞⎛

Fx1i

⎞

⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ Vp1i ⎠ = ⎝ − sin i cos i 0 ⎠ ⎝ Fy1i ⎠ Mp1i

0

0

(17)

Mz1i

1

Similarly, for pivot P2i , the displacement at the end of the pivot can be formulated as,







xt2i , yt2i , t2i =  Pp2i , Vp2i , Mp2i



(18)

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5

1

Item

Value

radius of the circumscribed circle of the fixed frame r1 radius of the circumscribed circle of the moving platform r2 the length of the intermediate links ld initially declination angle of the moving platform ␸p

53 mm 12.5 mm 80 mm ◦ −6.935   5  , ,  3 3

direction angle of the actuator ␣i

DISPLACEMENT STEP=61 SUB =1 TIME=61 DMX =22.8911

Table 2 Geometric and material parameters of the flexure pivot. Item

Value

Spring leaf length L Width of the leaf Thickness of the leaf Cross angle ␤ Young’s modulus of the material E Yield stress of the material ␴p

20 mm 10 mm 0.4 mm 45◦ 110 GPa 900 MPa

where the force acting on the pivot P2i in LCS Bi xy is

⎛

⎞

Pp2i

⎛



cos i + t1i





sin i + t1i



0

⎞⎛

Fx2i

Fig. 7. Finite element model of the CPM.

⎞

    ⎟⎜ ⎜ ⎟ ⎜ ⎟ ⎝ Vp2i ⎠ = ⎝ − sin i + t1i cos i + t1i 0 ⎠ ⎝ Fy2i ⎠ Mp2i

0

0

1

Mz2i (19)

Moreover, the static equilibrium equations of the whole manipulator expressed via F1i are given as below.

⎧ 3  ⎪ ⎪ ⎪ Fx1i = 0 ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ 3 ⎨

Fy1i = 0

⎪ ⎪ i=1 ⎪ ⎪ ⎪ 3 3 ⎪   ⎪   ⎪ ⎪ M + Fx1i (yCi − yP ) − Fy1i (xCi − xP ) = 0 ⎪ z1i ⎩ i=1

Fig. 8. Displacement of the actuators.

(20)

i=1

The nonlinear equations defined by Eqs. (14) and (20) contain 12 constraints which correspond to the number of unknowns i.e., the intermediate forces F1i , and the pose of the moving platform (xp ,yp ,ϕ) for the inverse kinematics (or the displacements of the actuators ␳i for the forward kinematics). Therefore, the kinematics of the CPM can be solved. A program is developed in MATLAB to solve those nonlinear equations. 4. Results and discussion In order to verify the NLKM for the 3-PRR CPM with flexure pivots proposed in this paper, FEA is carried out by utilizing the commercial software ANSYS. The detailed geometric parameters of the CPM are listed in Table 1. The Ti6Al4V alloy is selected as the material to fabricate the spring leaves for its large elastic limit, and the geometric parameters and the material properties of the flexure pivots are listed in Table 2. To improve the efficiency of the simulation, a planar finite element model of the CPM is established by applying the APDL language as shown in Fig. 7, in which all the parts are modelled as lines. The spring leaves in the flexure pivots are meshed by 100 elements of beam 189 which is suitable for large deflection analysis, and the geometric nonlinearity option is turned on. The other parts are modelled as rods with a radius of 10 mm and meshed into 20 elements, and the Young’s modulus for these rods

are 1000 times higher than the leaves, so the deformation of those parts can be neglected. 4.1. Inverse kinematic analysis To investigate the proposed inverse kinematics of the CPM, let the moving platform follow a given circle path with a radius of 20 mm, meanwhile, the rotation angle of the moving platform is changing form −3◦ to 3◦ . The 61 tested points are given as below.



⎧ ⎪ t x (ti ) = 20 cos ⎪ ⎪ 30 i ⎪ ⎪ ⎨

y (ti ) = 20 sin

⎪ ⎪ ⎪ ⎪ ⎪ ⎩

m

(ti ) =

 t 30 i

ti = 0, 1, ..., 60

(21)

t −3 10

The displacements of the actuators for inverse kinematic analysis calculated by the proposed nonlinear inverse kinematic model (NLIKM), the FEA and the conventional CKM are plotted respectively in Fig. 8. It can be seen that, all the displacement curves are sine shaped (with different phase delay) and the peak value is a little larger than the radius of the given path. To illustrate the differences of the results more clearly, Fig. 9 presents the actuators’ displacement errors of NLIKM and CKM compared with the FEA results. The maximum deviation of NLIKM compared with the FEA results is about 0.033 mm while the maximum deviation of CKM compared with the FEA results is 0.83 mm, which is more than 25 times larger than the former.

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Fig. 9. Displacement deviation of the actuators.

Fig. 11. Trajectory at the center of the moving platform.

Fig. 10. Rotation angles of the flexure pivots.

Fig. 12. Trajectory error of CKM and NLKM.

It also should be pointed out that, it took 134.4 s to obtain the results for the NLIKM on a personal computer (3.2 GHz Core i5-4460 processor and 6 GB RAM), while the time consumption for FEA by ANSYS is 342.217 s on the same calculation platform. It indicates that the NLKM is twice more efficient to predict the inverse kinematics of the CPM with large deformation, when compared with commercial software calculation. The corresponding rotation angles of the flexure pivot P1i obtained by NLIKM, CKM and FEA are also plotted respectively in Fig. 10. It can be seen that all the pivots needs to rotate from about −18.5◦ to 15.5◦ during the whole moving process, although there are some differences between the three flexure pivots. The maximum prediction error of the pivots’ rotation for NLIKM is 0.015◦ , but for the CKM, the maximum prediction error of the pivots’ rotation is 0.36◦ . The prediction error for CKM is much larger than NLIKM. Generally speaking, small deformation refers to the flexure joint rotate an angle less than 5◦ [27,29] and intermediate deformation refers to the flexure joint rotate an angle less than 10◦ [30,37]. In this paper, the deformation of the flexure pivots is apparently greater than intermediate range. In order to improve the accuracy of NLIKM or solve the kinematic behavior of the CPM in a larger deformation range, methods like the chained beam-constraint-model (CBCM) proposed in [35] can be used to model the deformation of flexure pivots, while the formulation of the nonlinear kinematic equations of the CPM do not need any change.

The maximum stress on the leaves for the three flexure pivots are 492 MPa, 479 MPa and 483 MPa respectively under their maximum rotation angle, which is far less than the yield stress of the material, i.e., which means the structure is safe enough to fulfill the designed path. And the rotation of pivots P2i can be easily obtained by applying Eq. (14). 4.2. Forward kinematic analysis The inverse kinematic model is used to calculate the displacements of the actuators to allow the moving platform follow the predetermined trajectories, thus the open loop performance of a CPM significantly depends on the accuracy of the kinematic model. To illustrate the influence of the deviation of actuators on the absolute accuracy of the CPM, the three group of displacements of the actuators obtained from the inverse kinematic analysis (FEA, NLIKM, CKM) are then employed as the inputs to the finite element model to conduct the forward kinematic analysis. The results are plotted in Figs. 11–13 , where FEA (NLIKM) means the FEA forward kinematics results by applying the actuators’ displacements from NLIKM, similarly, FEA (FEA) and FEA (CKM) are the forward kinematics results by using the actuators’ displacements from FEA and CKM respectively. Fig. 11 is the trajectory at the center of the moving platform, Figs. 12 and 13 are the trajectory errors and angular displacement

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Acknowledgements This work was supported by National Natural Science Foundation of China under Grant No. 51475113, Natural Science Foundation of Heilongjiang Province under Grant No. E2015006, the State Key Lab of Self-planned Project under Grant No. SKLRS201501A03, and the Programme of Introducing Talents of Discipline to Universities under Grant No. B07018.

References

Fig. 13. Angular error of CKM and NLKM.

errors of the moving platform. By comparison with FEA (FEA), the NLIKM causes a maximum trajectory error of the moving platform of 0.015 mm, and a maximum rotation error caused by NLIKM is 0.085◦ . The CKM causes a maximum deviation of the moving platform of 0.798 mm. These errors are much larger than the NLIKM. For the rotation angle, the result of the CKM is totally unacceptable, and a maximum prediction error is −4.379◦ which exceeds 150% of the given deformation angle. Moreover, to verify the nonlinear forward kinematic model (NLFKM) proposed in this paper, the actuators’ displacements obtained from FEA are also applied as inputs for the NLFKM, refer as NLFKM(FEA). The trajectory and rotation angle of the moving platform are compared with FEA(FEA). It can be seen from Fig. 12 and Fig. 13, the forward kinematic solution obtained by FEA and the NLFKM proposed in this paper agreed very well. The maximum trajectory error and angular error of the moving platform are 0.023 mm and 0.165◦ respectively. The NLFKM(FEA) spent 163.3 s to complete the forward kinematic calculation while FEA by ANSYS took 381.2s, i.e., the efficiency of the NLFKM is still more than twice the speed of the FEA calculation.

5. Conclusions In this paper, a kinetostatic modelling approach for a long stroke 3-PRR flexure pivot based CPM has been proposed to improve the open loop motion accuracy of the CPM. The model considers the parasitic motion of the flexure pivots during the deformation, and displacement constraint equations and static equilibrium equations of the CPM are established under the deformed configuration. Both inverse kinematic analysis and forward kinematic analysis can be conducted by the proposed methodology. Numerical results compared with FEA show that the proposed model can predict the deformation of the CPM precisely, and the calculation efficiency of the NLKM is more than twice as high as the FEA. The maximum prediction errors of the actuator displacements are 0.033 mm for inverse kinematic analysis, and the maximum prediction errors of the moving platform’s position and rotation are 0.023 mm and 0.165◦ respectively for forward kinematic analysis. By applying the NLKM, the prediction accuracy of the CPM’s kinematic behavior has been improved more than 20 times compared with the conventional method. Moreover, the concept and approach outlined in this paper are generic, and can be extended to the other type of flexure joint involved CPMs.

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Please cite this article in press as: Yang M, et al. Kinetostatic modelling of a 3-PRR planar compliant parallel manipulator with flexure pivots. Precis Eng (2017), http://dx.doi.org/10.1016/j.precisioneng.2017.01.002