A large-stroke compliant joint–based force/displacement integrated sensor

Abstract

The design methodology and validation of a compliant translational joint–based force/displacement integrated sensor is presented in this article. The stiffness analysis of the large displacement high precision compliant translational joint is developed, in which the screw theory–based symbolic formulation method for structures is employed. By combining the stiffness matrix of the single compliant beams and components in this joint, the entire stiffness matrix is derived. The stiffness matrix is validated by finite element analysis (FEA) method. Finally, the compliant translational joint was fabricated with a three-dimensional printer and equipped with a linear position sensor and microcontroller. The displacement of the translational joint is measured and then the force is calculated using the stiffness matrix. A calibration is conducted so that the sub-Newton precision of the sensor is achieved.

Keywords

Compliant joint flexure stiffness analysis screw theory force sensor

Introduction

In certain special applications such as redundantly actuated parallel manipulators as shown in Figure 1, the displacement/force integrated sensor is of great use for measuring both displacement and force simultaneously, and improving the safety of the mechanism in the presence of internal redundant actuating forces. A mechanical element or equipment with the ability to conduct displacement/force transmission is indispensable in this kind of application. Moreover, compared with the traditional mechanism, compliant mechanisms have significant characteristics including monolithic manufacturing, high precision, no need of lubrication, and no pollution.^1–3 And the compliant mechanism can also overcome the high cost of the processing and assembly of traditional mechanism, the clearance of the kinematic pair and the friction, and so on. Therefore, compliant mechanisms have found wide applications in the micro manipulation,⁴ medical instrumentation,⁵ precise positioning,⁶ micro-electromechanical systems (MEMS) devices,^7,8 and so on. A compliant joint (also known as flexure) is one of the basic components of the compliant mechanisms, which utilizes the small deformation and self-recovery characteristics of compliant elements to eliminate the backlash and mechanical friction, thus obtains ultra-high displacement resolution, and compact mechanical structure, high stiffness, and quick response.

Figure 1.

A displacement/force integrated sensor in a radar system.

As being subject to the constraints of the permissible stresses and strains in the material, the motion range of compliant joints is relatively small. When the yield stress of the material is reached, elastic deformation becomes plastic, after which the behavior of compliant joint is unstable and unpredictable. Therefore, the range of motion is determined by material property and geometry of the joint. Trease and Moon et al.⁹ proposed the configuration of a large displacement complaint translation joint (CTJ), which has a lot of advantages illustrated in their research, such as the large stroke, the small axial drift, the weak effect of stress concentration, the strong shaft stiffness, and the compact structure.

Once the CTJ is used as a transmission element from force to displacement, the computation of stiffness (or compliance) of the compliant joint is critical in the design process. The fundamental task of the stiffness analysis is the mathematical description of the load and deformation relation of the compliant joint. This article adopts the analytical method of compliance of the compliant joint based on the screw theory,^10–12 in which the stiffness matrices of multiple compliant components are combined, through the decomposition of the component of the complex compliant joint, and then the entire stiffness matrix of the compliant translation joint is formed. Finally, the effectiveness of the result is verified through the finite element analysis (FEA) simulation. Finally, a microcontroller-based measuring and displaying system is equipped on the CTJ to achieve the goal of force/displacement measurement.

The remaining of this article is organized as follows. The structure description of the large-stroke CTJ is given in section “Structure description of the large-stroke translational joint”; The detailed analysis of the stiffness matrix of the compliant joint is developed in section “Analysis of stiffness matrix of the CTJ”; The verification via the FEA method is carried out in section “Verification via the FEA”; The fabrication of and experiment on the compliant translational joint–based displacement/force sensor are elaborated in section “Fabrication and experiment”; As a summary, some meaningful conclusions are drawn in section “Conclusion.”

Structure description of the large-stroke translational joint

The compliant translation joint (CTJ) shown in Figure 2 is an entirety that is composed of two identical parts in quadrature as shown in Figure 3. The entire CTJ consists of 24 flexible beams in the middle and the rest six connecting rods regarded as rigid bodies. Components labeled with B and S are respectively connected to different parts, and the sliding motion between these two components is realized by the lateral deformation of the thin flexible beams.

Figure 2.

Large-stroke CTJ.

Figure 3.

Components topology in the CTJ.

Two hypotheses on the CTJ component are made firstly to derive the stiffness: (a) six connecting rods are ideally rigid and (b) 24 flexible beams satisfy the linear elastic model and the small deformation conditions in the operation of CTJ. These two identical parts in quadrature of the CTJ work together so that they are of parallel relation concerning stiffness. The entire stiffness matrix of the CTJ can be formed by the stiffness matrices of these two identical parts represented in the same coordinate system. To start with, therefore, the entire stiffness matrix of the vertical part shown in Figure 3 needs to be derived.

The four substructures (or parts) in the circles in Figure 3 comprise totally 12 flexible beams, and they also have a parallel relationship among the other three flexible beams in each circle. Both ends of these parts are fixed to the two rigid bodies. Four substructures labeled by 1, 2, 3, 4 in Figure 3 are constituted by the rigid body at both ends of the three flexible parts parallel. The substructure numbers are referred to hereinafter. The substructures in Figure 3 are connected by four rigid parts labeled U, D, B, S, respectively. Parallel substructures 1 and 2 are connected in series by the rigid body U, while parallel substructures 3 and 4 are connected in series by the rigid body D. Then the above-mentioned two serial substructures are connected in parallel by the rigid bodies S and B. Here the rigid body B is considered as the fixed reference base, while the rigid body S is the actuating body to be studied. Eventually, the relationship between the force screw and the displacement screw generated at the origin of a coordinate system on the rigid body S will be derived (i.e. the stiffness matrix or compliance matrix).

Analysis of stiffness matrix of the CTJ

Stiffness matrix of the parallel substructures 1 and 2 in O ₁ X ₁ Y ₁ Z ₁

For the parallel substructure 1 shown in Figure 4, one end of its flexible beams is fixed to the rigid body B, while the frame O ₁ X ₁ Y ₁ Z ₁ is established with the origin at the center of the other end (fixed to the rigid body U). In this case, the stiffness matrix of the flexible beams in O ₁ X ₁ Y ₁ Z ₁ is also the stiffness matrix at the center of the free end of cantilever beam. By the knowledge of mechanics of materials,¹³ one can obtain

(\begin{matrix} θ_{x} \\ θ_{y} \\ θ_{z} \\ δ_{x} \\ δ_{y} \\ δ_{z} \end{matrix}) = (\begin{matrix} 0 0 \frac{{L_{b}}^{2}}{2 E I_{x}} \frac{L_{b}}{E I_{x}} 0 0 \\ 0 0 0 0 \frac{L_{b}}{G I_{t}} 0 \\ \frac{- {L_{b}}^{2}}{2 E I_{z}} 0 0 0 0 \frac{L_{b}}{E I_{z}} \\ \frac{{L_{b}}^{3}}{3 E I_{z}} 0 0 0 0 \frac{- {L_{b}}^{2}}{2 E I_{z}} \\ 0 \frac{L_{b}}{EA} 0 0 0 0 \\ 0 0 \frac{{L_{b}}^{3}}{3 E I_{x}} \frac{{L_{b}}^{2}}{2 E I_{x}} 0 0 \end{matrix}) (\begin{matrix} F_{x} \\ F_{y} \\ F_{z} \\ M_{x} \\ M_{y} \\ M_{z} \end{matrix})

(1)

where the left vector ${(θ_{x}, θ_{y}, θ_{z}, δ_{x}, δ_{y}, δ_{z})}^{T}$ in equation (1) is the twist (displacement screw) at the origin, the right vector ${(F_{x}, F_{y}, F_{z}, M_{x}, M_{y}, M_{z})}^{T}$ is the wrench (force screw) at the origin, and this convention applies hereinafter. For simplicity, equation (1) is denoted as $T_{12} = C_{12} W_{12}$ ; $C_{12}$ is the compliance matrix of flexible intermediate portion of substructure 1. The first digit of 1 in the subscript here represents the parallel substructure 1, and the second digit of 2 represents the second flexible part from the left in this parallel structure. This subscript convention applies similarly hereinafter.

Figure 4.

Relationship of the CTJ compliant components.

The origins of local coordinate systems of the first and the third flexible parts in substructure 1 are located at the center of the end plane by which they are fixed to the rigid body U. Their orientations are identical to that of O ₁ X ₁ Y ₁ Z ₁. Therefore, the expression of either stiffness matrix of these two parts is the same as $C_{12}$ in their corresponding local coordinate systems. Thus, these three compliance matrices satisfy $C_{11} = C_{13} = C_{12}$ , and their corresponding stiffness matrices are $K_{11} = K_{13} = K_{12} = {C_{12}}^{- 1}$ . Furthermore, the local stiffness matrices of the first and third flexible beam of the substructure 1 from the left are transformed to $O_{1} X_{1} Y_{1} Z_{1}$ . And the coordinates transformation for the first flexible beam is $R_{11} = I_{3 \times 3}, {\overset{⇀}{d}}_{13} = {(\begin{matrix} - (L_{2} + t) & 0 & 0 \end{matrix})}^{T}$ . Then the transformation matrix is

A {d_{11}}^{- 1} = (\begin{matrix} {R_{11}}^{T} O \\ - {R_{11}}^{T} D_{11} {R_{11}}^{T} \end{matrix})

(2)

where $D_{11}$ is the skew-symmetric matrix of ${\overset{⇀}{d}}_{13}$ , while O is zero matrix and it is similar hereinafter.

The stiffness matrix $K_{11}$ is transformed into $K'_{11} = A d_{11} \cdot K_{11} \cdot A {d_{11}}^{- 1}$ in $O_{1} X_{1} Y_{1} Z_{1}$ , and the stiffness matrix $K_{13}$ is transformed into $K'_{13} = A d_{13} \cdot K_{13} \cdot A {d_{13}}^{- 1}$ in $O_{1} X_{1} Y_{1} Z_{1}$ at the same time, where

A d_{13} = (\begin{matrix} R_{13} O \\ D_{13} R_{13} R_{13} \end{matrix})

$R_{13} = I_{3 \times 3}$ , $D_{13}$ is the skew-symmetric matrix of $\begin{matrix} {\overset{⇀}{d}}_{13} = {(\begin{matrix} L_{2} + t & 0 & 0 \end{matrix})}^{T} \end{matrix}$ .

Since the stiffness matrix of the second flexible part is already described in the coordinate system of $O_{1} X_{1} Y_{1} Z_{1}$ , so one can obtain $K'_{12} = K_{12}$ . As a result, the entire stiffness matrix of the parallel substructure is $K_{1} = K'_{11} + K'_{12} + K'_{13}$ . In the similar way, the entire stiffness matrix of the parallel structure 2 is $K_{2} = K'_{21} + K'_{22} + K'_{23}$ .

Combinational stiffness matrix of serial structure 1 and 2 in $O_{2} X_{2} Y_{2} Z_{2}$

The compliance matrix of structure 1 is $C_{1} = {K_{1}}^{- 1}$ in the coordinates of $O_{1} X_{1} Y_{1} Z_{1}$ , and that of substructure 2 is $C_{2} = {K_{2}}^{- 1}$ in the coordinate of $O_{2} X_{2} Y_{2} Z_{2}$ . The coordinate transformation from $O_{1} X_{1} Y_{1} Z_{1}$ to $O_{2} X_{2} Y_{2} Z_{2}$ is ${}_{1}^{2}R = I_{3 \times 3}, {\overset{⇀}{d}}_{1 t 2} = {(\begin{matrix} - (3 t + 2 L_{2} + L_{3}) & L_{b} & 0 \end{matrix})}^{T}$ . Then, one can obtain

A d_{s 1} = (\begin{matrix} _{1}^{2} R O \\ D_{1 t 2}_{1}^{2} R_{1}^{2} R \end{matrix})

(3)

The compliance matrix of the parallel structure 1 transformed into O ₂ X ₂ Y ₂ Z ₂ is

C_{1 s} = A d_{s 1} \cdot C_{1} \cdot A {d_{s 1}}^{- 1}

(4)

As the compliance matrix of the parallel substructure 2 is expressed in the coordinate systems of $O_{2} X_{2} Y_{2} Z_{2}$ , $C_{2 s} = C_{2}$ . Then the entire compliance matrix obtained by the connection between the parallel substructures 1 and 2 in series is $C_{s 12} = C_{1 s} + C_{2 s}$ . So the entire stiffness matrix of the serial structures composed of 1 and 2 is $K_{s 12} = {C_{s 12}}^{- 1}$ .

Combinational stiffness matrix of serial substructures 1 and 2 in $O_{4} X_{4} Y_{4} Z_{4}$

Figure 4 shows the difference between the local coordinate system of $O_{3} X_{3} Y_{3} Z_{3}$ established in parallel substructure 3 and $O_{4} X_{4} Y_{4} Z_{4}$ in parallel structure 4. In Figure 4, the substructures 3 and 4 can be considered to be the rotation of substructures 1 and 2 around the horizontal axis. It is also noted that these two coordinates $O_{3} X_{3} Y_{3} Z_{3}$ and $O_{4} X_{4} Y_{4} Z_{4}$ , respectively, coincide with $O_{1} X_{1} Y_{1} Z_{1}$ and $O_{2} X_{2} Y_{2} Z_{2}$ .

By the property of the upper and lower symmetry of the CTJ, the entire stiffness matrix obtained by the connection of substructures 3 and 4 in series in the coordinates of $O_{4} X_{4} Y_{4} Z_{4}$ is by coincidence the same as the one obtained by the connection of the substructures 1 and 2 in series in the coordinates of $O_{2} X_{2} Y_{2} Z_{2}$ , namely $K_{s 34} = K_{s 12}$ holds for the stiffness matrix.

Combinational stiffness matrix of the compliant structure

As shown in Figures 4 and 5, the origin of the coordinate of $O_{v} X_{v} Y_{v} Z_{v}$ is located at the intermediate point between the second compliant part of the substructures 2 and 3. $O_{v} X_{v} Y_{v} Z_{v}$ is the very coordinate in which the combinational stiffness matrix of substructures is represented. Therefore, the $K_{s 12}$ and $K_{s 34}$ obtained in the last two sections should be transformed to $O_{v} X_{v} Y_{v} Z_{v}$ and then summed up.

Figure 5.

Compliant beams in the horizontal plane.

The transformation from the coordinate of $O_{2} X_{2} Y_{2} Z_{2}$ to $O_{v} X_{v} Y_{v} Z_{v}$ is

A d_{s 12} = (\begin{matrix} R_{s 12} O \\ D_{s 12} R_{s 12} R_{s 12} \end{matrix})

(5)

where $R_{s 12} = I_{3 \times 3}$ , $D_{s 12}$ is the skew-symmetric matrix of ${\overset{⇀}{d}}_{s 12} = {(\begin{matrix} 0 & w / 2 & 0 \end{matrix})}^{T}$ . Similarly, the transformation from the coordinates of $O_{4} X_{4} Y_{4} Z_{4}$ to $O_{v} X_{v} Y_{v} Z_{v}$ is

A d_{s 34} = (\begin{matrix} R_{s 34} O \\ D_{s 34} R_{s 34} R_{s 12} \end{matrix})

(6)

where

R_{s 34} = (\begin{matrix} 1 0 0 \\ 0 - 1 0 \\ 0 0 - 1 \end{matrix})

$D_{s 34}$ is the skew-symmetric matrix of ${\overset{⇀}{d}}_{s 34} = {(\begin{matrix} 0 & - w / 2 & 0 \end{matrix})}^{T}$ .

The combinational stiffness matrix converted from $K_{s 12}$ and $K_{s 34}$ in the coordinate system of $O_{v} X_{v} Y_{v} Z_{v}$ in Figure 5 is

\begin{matrix} K_{v} = K'_{s 12} + K'_{s 34} = A d_{s 12} \cdot K_{s 12} \cdot A {d_{s 12}}^{- 1} \\ + A d_{s 34} \cdot K_{s 34} \cdot A {d_{s 34}}^{- 1} \end{matrix}

(7)

Entire stiffness matrix of the compliant translational joint

Figure 3 is a schematic diagram of the vertical part of the compliant translational joint, which has derived the entire stiffness matrix in the coordinate of $O_{v} X_{v} Y_{v} Z_{v}$ . In view of the axial symmetry of this CTJ structure, the expression of the entire stiffness matrix of the horizontal part in the coordinates of $O_{h} X_{h} Y_{h} Z_{h}$ is equal to $K_{v}$ , that is, $K_{h} = K_{v}$ . However, in order to get the stiffness matrix of the entire compliant prismatic which is composed of the vertical and horizontal components, one can convert the horizontal stiffness matrix from the coordinate system of $O_{h} X_{h} Y_{h} Z_{h}$ to $O_{v} X_{v} Y_{v} Z_{v}$ and then adds it to the stiffness matrix of the vertical part. Eventually, one can obtain the entire stiffness matrix of the CTJ in the coordinates of $O_{v} X_{v} Y_{v} Z_{v}$ .

The transformation from the coordinates of $O_{h} X_{h} Y_{h} Z_{h}$ to $O_{v} X_{v} Y_{v} Z_{v}$ is

A d_{htv} = (\begin{matrix} R_{htv} O \\ D_{htv} R_{htv} R_{htv} \end{matrix})

(8)

where

R_{htv} = (\begin{matrix} 1 0 0 \\ 0 0 1 \\ 0 - 1 0 \end{matrix}), d_{htv} = {(\begin{matrix} 0 0 0 \end{matrix})}^{T}

(9)

The subscript htv indicates horizontal to vertical. Therefore, once $K_{h}$ is converted to the coordinates of $O_{v} X_{v} Y_{v} Z_{v}$ , it has the new form of $K'_{h} = A d_{htv} \cdot K_{h} \cdot A {d_{htv}}^{- 1}$ . As for $K_{v}$ , since it is already in the coordinate of $O_{v} X_{v} Y_{v} Z_{v}$ , the transformation matrix is an identity matrix, that is, $K'_{v} = K_{v}$ holds. The entire stiffness matrix of the CTJ represented in the coordinate of $O_{v} X_{v} Y_{v} Z_{v}$ takes the following formula

K_{CTJ} = {K_{h}}^{'} + {K_{v}}^{'}

(10)

where K_CTJ denotes the entire stiffness matrix of the CTJ represented in coordinate of $O_{v} X_{v} Y_{v} Z_{v}$ .

Verification via the FEA

In order to verify the effectiveness of the stiffness matrix of the CTJ under the assumption of the small deformation and the linear elastic material, the static analysis is made using finite element method with SolidWorks Simulation module. An comparison between the results from the analytical model and the FEA model is implemented. The following parameters and their values are adopted to build the compliant translational joint model in the FEA simulation (Table 1).

Table 1.

Values of the design parameters of the CTJ.

Parameter	Physical sense	Value	Unit
t	Thickness of the compliant beams	1	mm
L_b	Length of the compliant beams	30	mm
w	Width of the compliant beams	10	mm
L₁	Length of the end rod	10	mm
L₂	Gap between the adjacent beams	5	mm
L₃	Gap between the symmetrical substructures	80	mm

In the FEA simulation, the meshing grade was set as Finest as a result 2880 Quadratic hexahedron (Solid186) elements were generated. And the material is polylactic acid (PLA), whose elastic modulus is $E = 4000 MPa$ and shear modulus is $G = 150 MPa$ . One end of the CTJ finite element model is fixed, and the other end is applied on an axial pressure of 40 N. Here, small sizes are used for the left and the right ends of the protruding parts. This operation can be explained by the fact that fixed and the actuating ends of both sides together with the four floating parts connecting each flexible part are all rigid in the analytical model. In fact, the entire virtual model constructed in SolidWorks is made from the same material, namely the entire model is considered as a flexible body; therefore, the deformation obtained in the FEA simulation will be a little greater than that in analytical model when bearing the same load. This particular operation in modeling process will weaken the errors from the numerical approaches.

In order to validate the aforementioned analytical stiffness result, L ₁ (illustrated in Figure 3) is decreased purposely here. The smaller size will yield the smaller deformation of the both end rods and make analytical model and FEA model more consistent to each other. It is also noted that this extra operation is particularly for model validation in this research, and it is thus no need to put in more consideration in practice for the following two reasons. Firstly, the assumption that some of the parts are rigid in this article has a small effect on the deformation of the flexible prismatic in practice; Secondly, the theoretical error caused by the small deformation and elastic material hypothesis and the error introduced by the rigid hypothesis in the formula will be compensated for by the immediately following sensor calibration altogether.

The deformation ratio for displaying is set to 50 and displacement plot along the axial direction is shown in Figure 6. From this figure, one can observe that the maximum displacement (displacement at the end plane where the axial loading pressure is applied) is 8.7734 mm.

Figure 6.

Axial displacement under the 40 N load.

Figure 7 shows that the largest von Mises stress in the compliant translational joint under the axial pressure of 40 N is $3.11 \times 10^{7} Pa$ , which is far less than the yield strength of $2.07 \times 10^{8} Pa$ for this kind of material. Therefore, the CTJ is still in the safe range of the linear elastic deformation under the axial loading pressure, so the results of deformation and displacement obtained by the simulation make sense. $K_{w} [1, 4]$ indicates the element in the first row and the fourth column of the stiffness matrix. With $K_{w} [1, 4]$ , one can calculate the relative displacement of 8.7602 mm between one end and the other end of the CTJ along the axial direction under the axial load of $F_{x} = 40 N$ . As seen, it is slightly less than the previous simulated results, and the ratio of the difference between the analytical and FEA simulated results is merely 0.15%. So the relative deviation is quite small and within the acceptable range. The cause accounting for the relative deviation is that the analytical result is based on the compliant CTJ, which is composed of flexible beams interconnected by some rigid parts, while the entire CTJ in FEA model is the identical material with an finite elastic modulus. Besides, the modeling methods between these two models are to some extent different and the stiffness matrix derived analytically is a linear approximation model within a small range. So the relative deviation in the comparison is introduced due to these combined causes.

Figure 7.

Von Mises stress under the 40 N load.

Additionally, through the entire stiffness matrix of the CTJ obtained above, one can obtain

F_{x} = K_{w} [1, 4] \cdot δ_{x}

(11)

In fact, this result agrees with our intuition and can be explained as follows. Under the loading force F perpendicular to a beam axis at the free end, the deflection of a cantilever beam is $3 EI / F L^{3}$ at the fixed point.¹⁴ So its equivalent stiffness is $3 EI / L^{3}$ . The stiffness matrix of the total 24 cantilever beams in parallel is thus $72 EI / L^{3}$ that is consistent to the results derived above.

Fabrication and experiment

To fabricate the prototype of this research, a MakerBot Replicator 2 model 3D printer with PLA material was employed. This 3D printer has a printing layer resolution of 0.1 mm. The PLA material has a density of 1250 kg/m³ and Young’s modulus of about 400 MPa from the data sheet. The measurement of displacement is conducted with a KSP-5 mm round self-recovery linear displacement sensor, which has a displacement range of 3 mm and repeatability of 0.01 mm. A BST-V51 microcontroller kit is used to read the displacement value, calculate the force, and display the result. The microcontroller is STC89C52 with main frequency of 11.0592 MHz. The on-board 8-bit A/D converter is utilized to implement the conversion of analog signal of voltage to digital signal.

After the fabrication and setting up of the force/displacement integrated senor. The calibration of the sensor was conducted as follows. A series of loads were applied on the vertically mounted sensor, the displacement of the sensor was recorded from the readings of the liquid crystal display. The force and displacement data pairs were recorded, so that the stiffness curve from linear data fitting was shown in Figure 8.

Figure 8.

Axial stiffness curve fitting of CTJ.

It is also noted that the total stiffness matrix of the integrated displacement/force sensor includes the stiffness of linear displacement sensor $k_{eq} = 0.2253 N / mm$ . When the calibration is conducted, based on the experimental result, the equivalent stiffness of the integrated sensor is computed.

Eventually, the equivalent value of stiffness $k_{eq} = 6.731 N / mm$ was written in the code of the microcontroller, and the force/displacement sensor can work smoothly, as shown in Figure 9. It is noted that from the equivalent stiffness of the CTJ, one can calculate Young’s modulus E = 5916 MPa with equation (11). This value falls into the rational range of the material of PLA.¹⁵ The heating and injecting of PLA material during the 3D printing reconstruct the material, so its material property is no longer the theoretical value on data sheet. The calibration of this force/displacement sensor eliminates the perturbation due to the straight substitution of material property in data sheet and also compensates the stiffness of the linear displacement senor.

Figure 9.

Measurement of the integrated sensor.

Conclusion

This article elaborates the development and verification of a large-stroke compliant translational joint–based force/displacement integrated sensor. The analytical stiffness matrix of the compliant translational joint is derived with screw theory. Then the sensor is implemented with a microcontroller and achieved the design specifications. Some meaningful conclusions can be drawn as follows:

Based on the assumption of small deformation for the compliant components, the stiffness matrix of the compliant joint can be analytically formulated using screw theory.

From the result obtained in FEA method, the derived stiffness matrix is of high accuracy and can be effectively applied to the stiffness analysis and synthesis of this kind of compliant joints.

In terms of inaccuracy of the material property of the prototype and other electronic equipments, the calibration of the integrated sensor on the stiffness should be conducted after fabrication. The performance out of the linear range of the force/displacement sensor can also be calibrated by experiment and accomplish the desired precision.

As a technical alternative solution, it is possible to use force sensor instead for the force/displacement measuring of the proposed integrated sensor. That is the future work by the authors, in which the precision, range, and response rapidity between these two types of technical solutions will be compared.

Footnotes

Academic Editor: Pedro AR Rosa

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China under Grant Nos 51405362 and 51490660, and “111 project” under Grants No. B14042. The authors would also like to appreciate the Editor, Associate Editors, and the reviewers for their valuable comments and suggestions.

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