Kinetostatic modeling and characterization of compliant mechanisms containing flexible beams of variable effective length

Mechanism and Machine Theory 147 (2020) 103770

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory

Research paper

Kinetostatic modeling and characterization of compliant mechanisms containing flexible beams of variable effective length Fulei Ma a, Guimin Chen b,∗ a

School of Electro-Mechanical Engineering, Xidian University, Xi’an, Shaanxi 710071, China State Key Laboratory for Manufacturing Systems Engineering and Shaanxi Key Lab of Intelligent Robots, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

b

a r t i c l e

i n f o

Article history: Received 14 August 2019 Revised 27 November 2019 Accepted 16 December 2019

Keywords: Compliant mechanism Variable effective length Kinetostatic modeling

a b s t r a c t Flexible beams of variable effective length, serve both transmission and actuation functionalities in compliant mechanisms, have been employed in many devices, e.g., thermal actuators and continuum robots. This difunctional feature is favorable when utilized for operations in confined space. However, the length variation introduces modeling difficulties, which poses a new challenge to designers. To accurate model flexible beams of variable effective length and characterize the behaviors of associated compliant mechanisms form the primary objectives of this study. The chained beam constraint model is revisited and extended to model beams of variable effective length. Modeling of a chevron shape thermal in-plane microactuator and a continuum mechanism are provided to demonstrate the effectiveness of the proposed method. The predicted results have a high degree of accuracy as compared to experimental results and nonlinear finite element results. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Compliant mechanisms employing flexible beams with variable effective length exhibit unique behaviors that could be useful for operations in restricted space [1]. In such compliant mechanisms, the flexible beams not only provide the required deflections for motion but also offer actuation functionality. This kind of compliant mechanisms can find their embodiments in actuators and continuum robotics, for example, Thermal In-plane Microactuators (TIMs) [2–4], continuum robots [5], zipper hinges [6], as depicted in Fig. 1. Two different TIMs, as shown Fig. 1(a) [3] and (c) [4], output a stroke or force through the length change of the hot legs induced by thermal expansion. Because they are simple in structure and easy to be fabricated, they have been integrated in many micro-systems as actuators, e.g., optical mechanical switches, bistable mechanical switches and micro-positioners [7]. For the continuum mechanism (which could be used to construct continuum robots [5,8–9]) shown in Fig. 1(b), the end effector can vary its position and orientation by changing the effective length of its flexible beams through pulling or dragging. The electrostatic actuated zipper hinge shown in Fig. 1(d) was utilized in self-folding structures [6]. It should be noted that a beam with variable length in the current work is limited to length change caused by active actuation (e.g., thermal expansion), while length change induced by axial stretching/compressing [10] is not included. ∗

Corresponding author. E-mail address: [email protected] (G. Chen).

https://doi.org/10.1016/j.mechmachtheory.2019.103770 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

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F. Ma and G. Chen / Mechanism and Machine Theory 147 (2020) 103770

Fig. 1. Illustration of compliant mechanisms actuated by beams of variable length. (a) A chevron shape Thermal-in-plane Microactuator (TIM). (b) A continuum mechanism in plane continuum robots. (c) A U shape TIM (d) A electrostatic actuated zipper hinge.

Flexible beams of variable length serve both transmission and actuation functionalities in compliant mechanisms, which always lead to highly compact designs. However, the length variation may introduce strong nonlinearities in the kinetostatic behaviors, which poses a new challenge to the modeling and design of such mechanisms. To the best of our knowledge, few work has been done on modeling flexible beams of variable length, most of which is focused on capturing the kinetostatic behaviors of a specific mechanism, i.e., the TIM. For example, Enikov et al. [11] presented an analytical solution employing the first-order approximation of the nonlinear strain-displacement relation for the axial deflection, which is suitable for beams undergoing intermediate deflections [12–13] but could yield large errors for large deflections; the pseudo-rigid-body model (PRBM) in Ref. [14] provides an effective way for analyzing TIM, but suffers from inaccuracies when modeling large deflections with inflection points; the elliptic integral solutions [15–16] give accurate predictions for kinetostatic behaviors of TIM, but are sensitive to the initial guess and may suffer from poor convergence rates, especially when multiple inflection points produced on the deflected beams. There still exists a need for a method that is robust and can be applied to a variety of compliant mechanisms containing flexible beams of variable effective length. The main objective of this work is to develop a general method to fully understand the nonlinear deflections of flexible beams of variable length and to accurately capture the kinetostatic characteristics of compliant mechanisms containing this kind of flexible beams. The rest of this paper is organized as follows. The chained beam constraint model [17–18] is revisited and extended to model beams of variable length in Section 2. The method is demonstrated by modeling the kinetostatics of two compliant mechanisms containing flexible beams of variable length in Section 3, and the results are discussed in Section 4. Finally, concluding remarks are made in Section 5.

2. Modeling of beams of variable effective length Fig. 2 shows a cantilever beam subject to transverse force Fo , axial force Po and moment Mo at its free end, resulting a deflection of tip coordinates Xo and Yo and end slope θ o . The geometric parameters of the beam are given as: length L, thickness T and width W. The variation of the effective length of the beam is denoted as Lα . Lα is a variable and usually the input of the mechanism. The length variation induced by axial stretching/compressing is not included in Lα . Similar to CBCM, the beam is equally discretized into N elements.

F. Ma and G. Chen / Mechanism and Machine Theory 147 (2020) 103770

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Fig. 2. Discretization of the flexible beam of variable length. (a) the beam is discretized into N elements (b) diagram of the i th element.

In order to incorporate the variation of the effective length and to eliminate the number of the parameters, we introduce a new normalization scheme:

Fo (L + Lα ) Po (L + Lα ) Mo ( L + Lα ) ; po = ; mo = NEI N 2 EI N 2 EI N Xo NYo xo = ; yo = ; θo = θo ( L + Lα ) ( L + Lα ) 2

2

fo =

(1)

For the i th element, its local coordinate frame (Oi Xi Yi ) is attached to and moves along with the free end of (i − 1)-th element, i.e., Oi . We use Pi , Fi and Mi for the transverse force, the axial force and the end moment applied on the free end of this element, all measured with respect to Oi Xi Yi , and i , i and α i the corresponding axial and transverse deflections and the end slope, respectively. The discretization introduces 6 N intermediate parameters: the deflection and load parameters of each element, all of which are normalized with respect to the effective length of an element (L + Lα )/N as

Pi (L + Lα ) Fi (L + Lα ) Mi ( L + Lα ) ; fi = ; mi = NEI N 2 EI N 2 EI N i N i = ; δi = ; αi = αi L + Lα L + Lα 2

2

pi =

λi

(2)

This normalization scheme not only guarantees the consistency of the proposed method with the framework of CBCM, but also effectively extends the capability of CBCM for modeling variable length beams. The force-deflection relations of each element are formulated using the beam constraint model (BCM) as [12,19]:

 



        pi 36 pi 2 −9 4.5 δi −3 δi δi + + αi 4 αi 11 αi 30 −3 6300 4.5        36 −3 δi  −9 4.5 δi 1  pi  t 2 pi δi αi δi αi − − λi = −3 4 αi 4.5 11 αi 12 60 6300 −6 4

fi 12 = mi −6

(3)

where t = T/(L + Lα ) is the normalized thickness with respect to the effective beam length. The static equilibrium equations at each discretization point can be written as:

⎡

cos θi ⎣ − sin θi (1 + λi )

sin θi cos θi −δyi

⎤⎡ ⎤

⎡

⎤

0 fi f1 0⎦ ⎣ p i ⎦ = ⎣ p 1 ⎦ 1 mi mi−1

(4)

where θ i is the tip angle of i th element’s coordinate frame with respect to the global coordinate frame and can be directly expressed by the unknowns:

θ1 = 0 , θi =

i−1

k=1

αk (i = 2, 3 · · · N )

(5)

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F. Ma and G. Chen / Mechanism and Machine Theory 147 (2020) 103770

Fig. 3. Diagram of a hot leg in TIM.

The geometric constraint equations of the beam of variable length can be written as (note that the parameters are normalized with respect to the effective length (L + Lα )):

⎧ N  ⎪ ⎪ ⎪ [(1 + λi ) cos θi − δi sin θi ] = xo ⎪ ⎪ i=1 ⎪ ⎨ N [(1 + λi ) sin θi − δi cos θi ] = yo ⎪ i=1 ⎪ ⎪ ⎪ N ⎪ ⎪ ⎩ αi = θo

(6)

i=1

The CBCM equations given in Eqs. (3-6) contain (6N+3) equations. Among the 6 parameters, i.e., the tip loads po , fo , and mo and the tip deflections xo , yo and θ o , given any 3 parameters, the other 3 can be obtained by numerically solving the CBCM equations. The model can be simply denoted as

( fo po mo xo yo θo N ) = 0

(7)

It should be noted that Lα is not explicitly included in the formulation but is incorporated in the model through normalization. This treatment yields a more generalized model that can be reduced to the original CBCM [17,18] when Lα is zero. 3. Examples Two mechanisms, including a chevron shaped TIM and a continuum mechanism (whose flexible beams are of variable effective length), are presented to illustrate the use of the method for kinetostatic modeling. 3.1. Chevron shaped TIM Fig. 3 shows the diagram of a chevron shaped TIM. Several pairs of hot legs sharing the same geometric parameters are placed symmetrically with respect to the center line. The legs are initially straight and have a slant angle of β with respect to the horizontal line. The number of the leg pairs is denoted as Nt (Nt = 3 for the TIM in Fig. 3). The geometric parameters of each hot leg include length L, thickness T and width W. The Young’s modulus of the material is denoted as E. Due to the symmetry of the TIM, we only take one hot leg into consideration. Establish the global coordinate frame XOY with its origin O placed at the fixed end of the leg while its X-axis placed along with the length direction of the leg. When a current is applied through the leg, its length varies due to thermal expansion (the length change is denoted as Lα ) which results in a stroke Y in the actuation direction. The hot leg can be modeled as fixed-guided beams with length of (L + Lα ): the tip rotation and horizontal displacement of the hot leg are constrained thus are zeros and the vertical tip force is zero when no external load is applied. These three conditions can be used in the place of the 3 given parameters to solve for the

F. Ma and G. Chen / Mechanism and Machine Theory 147 (2020) 103770

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Fig. 4. Diagram of the continuum mechanism.

vertical displacement, the tip moment and the horizontal force. The normalized tip deflections of the leg are defined as:

N Xo N (L + Y sin β ) = L + Lα L + Lα NYo NY cos β yo = = L + Lα L + Lα θo = 0 xo =

(8)

The load-deflection equations are simply denoted using CBCM as:

( fo po mo xo yo 0 N ) = 0

(9)

where fo , po and mo are the normalized tip loads of the beam. By using the equilibrium relation on the shuttle, the output force of the TIM Ft can be formulated as:



Ft = 2Nt

N 2 foEI

(L + Lα )2

cos β +

N 2 poEI

(L + Lα )2



sin β

(10)

3.2. Continuum mechanism Fig. 4 shows the diagram of a continuum mechanism in its deflection configuration. Two flexible beams are placed parallelly with their right ends rigidly connected to an end effector whose length is Lb . The initial lengths of the two beams are L1 and L2 , while their thicknesses and widths are identical, denoted as T and W, respectively. E is the Young’s modulus of the material. An input force F is applied at the end of beam 2 to change its effective length. It should be noted that the sign of Lα can be negative, which means the effective length of the beam is reduced. As shown in Fig. 4, we place the origin of the local coordinate frame for beam 1 (O1 ×1 Y1 ) at its fixed end O1 , while the origin of the local coordinate frame for beam 2 (X2 O2 Y2 ) at its fixed end O2 . The tip loads of beam 1 with respect to its local coordinate frame are denoted as axial force Po1 , transverse force Fo1 and end moment Mo1 , and the tip deflections are denoted by Xo1 , Yo1 and θ o1 , respectively. Similarly, the tip loads and deflections of beam flexure 2 with respect to X2 O2 Y2 are denoted as Fo2 , Po2 , Mo2 , Xo1 , Yo1 , and θ o1 . Two beams are divided into N1 and N2 elements, respectively, and the deflection parameters and the load parameters are nondimensionalized as:

fo1 = xo1

Fo1 L1 2 2

; po1 =

Po1 L1 2

N1 EI N1 2 EI N1 Xo1 N1Yo1 = ; yo1 = L1 L1 Fo2 (L2 + Lα )

2

fo2 = xo2

; mo1 =

Mo1 L1 ; N1 EI

Po2 (L2 + Lα )

2

; po2 =

N2 2 EI N2 2 EI N2 Xo2 N2Yo2 = ;y = (L2 + Lα ) o2 (L2 + Lα )

; mo2 =

Mo2 (L2 + Lα ) ; N2 EI (11)

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F. Ma and G. Chen / Mechanism and Machine Theory 147 (2020) 103770 Table 1 Parameters of the TIM. Parameters

E

L

W

T

β

Design I

162 GPa

300 μm

6.95μm

3.45μm

0.5°

Fig. 5. Force-displacement relationship and stiffness of the TIM.

The load-deflection equations for the two beams are simply represented by:

1 ( fo1 po1 mo1 xo1 yo1 θo1 N1 ) = 0 2 ( fo2 po2 mo2 xo2 yo2 θo2 N2 ) = 0

(12)

The loop-closure equations are given as:

Xo2 − Xo1 = Lb sin θo1 Yo2 − Yo1 = Lb (1 − cos θo1 )

(13)

θo1 = θo2 Applying static balancing to Lb yields:

⎧ N2 f EI 1 o1 + ⎪ ⎪ L2 ⎪ ⎨ 1 N12 po1 EI L21

+

N22 fo2 EI

(L2 +Lα )2 N22 po2 EI

+ Ft = 0 =0

( L2 + Lα ) ⎪   ⎪ ⎪ ⎩ N1 mo1 EI + N2 mo2 EI + N22 fo2 EI2 + Ft Lb sin θo1 + L1 L2 + Lα 2 ( L +L ) 2

2

α

(14) N22 po2 EI

L cos θo1 (L2 +Lα )2 b

=0

The input force can be calculated as:

F=

N2 2 fo2 EI

( L 2 + L α )2

(15)

4. Results and discussions 4.1. TIM This example is taken from Ref. [20]. The TIM was fabricated using the SUMMiT VTM surface micromachining process. A force gauge and a vernier were employed to conduct the in situ measurement of the force and displacement, respectively. The parameters of the TIM are listed in Table 1, which are identical to those used in Ref. [20]. Suppose a current of 11.13 mA is applied, and the length of each hot leg is increased by Lα = 0.15 μm (predicted by the finite difference method [21,22]). The load-displacement curve of the TIM predicted by the kinetostatic model is plotted in Fig. 5. The experimental results obtained in Ref. [20] are also plotted in this figure for the purpose of comparison. The results show a good agreement, which indicates the effectiveness of the kinetostatic model.

F. Ma and G. Chen / Mechanism and Machine Theory 147 (2020) 103770

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Fig. 6. Characteristics of TIM at different Lα . Table 2 Parameters of the continuum mechanism. Parameters

E

L1

L2

Lb

W

T

Design I

200 GPa

60 mm

60 mm

30 mm

10 mm

0.25 mm

The results show that the TIM reaches its maximum stroke of 5.5 μm (no-load displacement, i.e., the applied force Ft = 0) for Lα = 0.15 μm. The output force Ft of the TIM increases as the displacement decreases. The stiffness of the TIM at different deflected positions can be obtained by differentiating the output force Ft with respect to Y, as shown in Fig. 5. We can see that the stiffness of the TIM decreases as the decease of Y, and when Y reaches point Q, the stiffness of the TIM becomes zero and then turns negative where the TIM is incapable of pushing. This critical point corresponds to the switch of the deflection mode of the hot legs from its first bending mode to the second bending mode [23,24]. For different effective length (Lα from 0.08 μm to 0.23 μm), the characteristics including the maximum stroke, the block force and the effective actuation range (defined as the range between the maximum stroke point and the critical point Q) are plotted in Fig. 6. An approximate linear relationship between the maximum stroke and Lα can be observed. The effective actuation range of the TIM at different Lα remains unchanged. Interestingly, with the increase of Lα , the block force of the TIM linearly increases at the beginning of the curve, while stays unchanged when Lα exceeds 0.14 um, which could be useful where a constant output force is required. 4.2. A continuum mechanism The parameters of the continuum mechanism are listed in Table 2. By applying a gradually increased displacement Lα , the rotation angle of the end effector predicted by the kinetostatic model is plotted in Fig. 7, as well as the required input force F. An FEA model of the continuum mechanism is built in ABAQUS with the flexible beam being meshed into 200 elements using quadric beam element B22 and the geometric nonlinearity option turned on. The results are plotted in Fig. 7 for comparison. The results obtained by both methods show a good consistency. An approximate linear relationship between θ and Lα is observed from the figure, which can facilitate the control of the continuum mechanism. We also find that a larger actuation force is required for dragging (i.e., the sign of the force is minus) than it for pushing. A continuum mechanism prototype is shown in Fig. 8. One of flexible beam end is rigidly fixed to ground, while the other serves as the input and is rigidly attached to a ball screw, on which a force gauge is mounted to record the input force. The ball screw is driven by a stepper motor to incrementally deflect the continuum mechanism. As the deflection of the mechanism, a diffraction grating scale (resolution of 20 μm) was used to measure the displacement and a force gauge (resolution of 0.05 N with 0.1% linearity over the measurement range ±100 N) to record the force. The results (including

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F. Ma and G. Chen / Mechanism and Machine Theory 147 (2020) 103770

Fig. 7. Variation of the rotation angle of the end effector θ and the input force F with the displacement Lα .

Fig. 8. Experimental setup for the continuum mechanism. Three configurations of the continuum mechanism for input forces (dragging, no force, and pushing, respectively).

θ and F) were recorded every 1 mm. The results are overlaid on those obtained the kinetostatic model for the purpose of comparison, as shown in Fig. 7. A good coincidence is observed between the experimental results and the model predictions, which verifies the effectiveness of the proposed model. For the cases where the friction in the sliding guide cannot be neglected, it can be easily incorporated into the kinetostatic model by using the following expression:

F=

N2 2 fo2 EI

(L2 + Lα )2

+ Fr

(16)

F. Ma and G. Chen / Mechanism and Machine Theory 147 (2020) 103770

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Fig. 9. Stiffness of the mechanism at different input Lα .

Fig. 10. Deflected Configurations of the mechanism at different Lα .

where Fr is the equivalent friction force. For the prototype, the comparison of the results indicates that the friction force can be neglected. To further investigate the stiffness characteristics of the continuum mechanism, the stiffness of the mechanism is obtained at different input displacement Lα . The results are plotted in Fig. 9. The deflected configurations at different Lα are depicted in Fig. 10 to show the dexterity of the mechanism.

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F. Ma and G. Chen / Mechanism and Machine Theory 147 (2020) 103770

5. Conclusions CBCM was extended for kinetostatic modeling of compliant mechanisms containing flexible beams of variable length. Two case studies including a TIM and a continuum mechanism were conducted to demonstrate the proposed method. The results show that the proposed method can accurately capture the nonlinear kinetostatic behaviors of such compliant mechanisms, and some interesting stiffness characteristics were observed as well. Declaration of Competing Interest The authors declare no conflict of interest in this article Acknowledgement The authors gratefully acknowledge the financial support from the Open Fund of State Key Laboratory of Robotics and System (HIT) under Grant No. SKLRS-2019-KF-07 and the National Natural Science Foundation of China under Grant Nos. 51805397, 51675396 and U1913213. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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