1 Introduction

Robots are increasingly adopted for workpieces deburring and polishing to overcome the problems of low production efficiency, high cost and poor consistency in manual polishing [1,2,3]. As pointed out in previous research, the contact force between the grinding tool and workpiece is the key factor that affects the polishing quality [4,5,6]. Excessive or insufficient contact force would greatly impair the quality of the surface roughness. Thus, it becomes an urgent issue to maintain the constant contact force in the grinding process to ensure the polishing quality and consistency [7,8,9].

Generally speaking, the constant force can be realized by two methods [10]. The first way is an active compliance force control. The polishing force can be controlled by a closed-loop controller. In addition, the robot can actively adapt to the changing of the machined surface to ensure the constant contact force [11, 12]. For instance, Masoud et al. proposed a resultant force control strategy for light abrasive surface grinding. The performance of resultant force control strategy with real-time tool deflection compensation algorithm is validated by the grinding experimental studies [13]. Wang et al. designed a vibration suppression method for large thin-walled shell grinding based on force control algorithm. The experimental results show that the grinding depth error is less than ± 0.1 mm, and the polishing quality of the workpiece surface can be significantly improved [14]. Xie proposed a method for grinding blade free-form surface based on PSO algorithm. The impedance controller based on PSO algorithm can realize the function of impedance parameters adjustment and the contact force between workpiece and grinding tool can be accurately controlled [15]. Xu put forward a zero drift and gravity compensation algorithm for robot belt grinding with complex geometry which combines force-position hybrid control strategy with PI controller to maintain the stability of force control during robot grinding process [16]. Although constant contact force can be realized by aforementioned methods, needing complex controller algorithm and expensive sensors limits their further applications. Moreover, the positioning accuracy would degrade the force control performance through the force-position hybrid control method which was used to realize active compliance force control [17].

In contrast, another method is the passive compliance force control which uses rigidity of the robot and the flexibility of the force controllable end-effector to realize force control. The path planning is realized by the robotic arm and the control of force is realized by the end-effector respectively [18, 19]. Mohammad proposed a force controllable end-effector for automatic deburring, which can reduce the inertial effect of polishing motor and spindle driven by micro driver [20]. Chen proposed an end-effector, including gravity compensation force controller and two new eddy current dampers, which were integrated into the end-effector to improve the dynamic performance of the system and suppress the vibration during grinding and polishing [21]. Liu proposed a polishing mechanism with characteristics of simple structure, small mass and good compliance performance. In addition, hardware circuit construction and force control algorithm design are modeled and analyzed detailedly [22]. Tang proposed a multi degree of freedom flexible terminal actuator for mould polishing robot. The buffer spring ensures the working pressure between the grinding head and the workpiece changes in a constant range [23]. The key component of the force controllable end-effector is the constant force compensation device. At present, the commonly used constant force compensation devices are type of electromagnetic, air bag and linear motor [24].

However, the above constant force compensation devices possess the disadvantages of low precision, slow response and poor environmental adaptability. In view of these shortcomings, a novel spatial CFM for grinding operation based on distributed flexure beam is proposed in this paper. The remainder of this paper is organized as follows: the design concepts of proposed CFM is depicted in Section 2; Section 3 establishes the mathematical model of the whole CFM and obtains the relationship between output force and input displacement; ANSYS Workbench is utilized to verify correctness of the theoretical analysis in Section 4; To eliminate the force variation, the PSO method is adopted to optimize the CFM in Section 5; In addition, experimental studies are conducted in Section 6 to verify the performance of the CFM; Finally, conclusions are made in Section 7.

2 Design concepts

CFM is a passive constant force system, which can provide a constant output force within a certain range of deformation relying on the characteristics of the structure itself. Compared with the active constant force system, the CFM can reduce the use of expensive sensors and complex control algorithms, thus reducing the cost.

To realize a certain constant force output, the combination of positive stiffness mechanism with negative stiffness mechanism is a typical method. Generally, the straight beam is taken as positive stiffness mechanism to neutralize negative stiffness region of bi-stable beam mechanism. However, the traditional symmetrical straight beam mechanism cannot take full advantage of negative stiffness region of the bi-stable beam mechanism. To overcome this issue, a novel type of folding beam as a positive stiffness mechanism is designed for the purpose of increasing the constant force travel range. The force-displacement relationship comparison of designed positive stiffness mechanism and straight beam mechanism is depicted in Fig. 1, which shows that the folding beam mechanism has a good linearity in long range.

Fig. 1
figure 1

Comparison of two kinds of positive stiffness mechanism. (a) CAD model. (b) Force-displacement curve comparison

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The autonomous polishing scenario using CFM system is shown in Fig. 2, the end-effector including voice coil motor (VCM), CFM and grinding head are installed at the end of robot arm through the connecting flange. When the VCM works, it drives the grinding head to approach the workpiece. As an intermediate part, the CFM placed between the motor unit and grinding tool which mounted on the CFM. Due to the constant force output characteristics of the CFM, the constant force can be maintained between grinding head and the workpiece in a certain range of motion, so as to finish the precision grinding and polishing operations.

Fig. 2
figure 2

The autonomous polishing scenario using CFM end-effector

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3 Analytical model of designed CFM

In this part, the analytical model of bi-stable beam and folding beam is established respectively. Then, the relationship between input force and output displacement of the CFM is derived. As shown in Fig. 2, the CFM is composed of three branched chains and a central column, each chain composed of a parallel bi-stable beams and a folding beam mechanism. Considering the identical architecture of each branch, an arbitrary chain is selected for analysis.

3.1 Modeling of bi-stable beam

As shown in Fig. 3, buckling and deformation of the bistable beam is resulted by the loading force Fn. In this figure, L denotes the length of the beam, 𝜃 represents the inclination angle between the bi-stable beam and the X axis, and Δxb is the displacement of the bistable beam under the force of Fn along Y direction. When the force Fn loading on the bistable beam exceeds the critical force of the material, the buckling force of the ith critical point of the guide beam can be obtained referring to the previous studies [25, 26]:

$$ F_{n} = \frac{4EI(\lambda_{n})^{2}}{L^{2}} $$
(1)

where λn = π,4.493,2π,...,n = 1,2,3,...

Fig. 3
figure 3

Buckling deformation of bi-stable beam mechanism

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Therefore, the negative stiffness of bistable beam kb can be expressed as follow:

$$ k_{b} \thickapprox \frac{33EI}{L^{3}} $$
(2)

where E represents the elastic modulus of the material, I = bd3/12 is the inertia moment of the guide beam, d is the thickness of the beam, and b is the width of the beam. Referring to Eqs. (1) and (2), the force-displacement relationship of the bistable beam can be written as:

$$ F_{n} = ES\frac{{{\varDelta}} x_{b}}{L}\left( \frac{{{\varDelta}} x_{b}}{L} - \sin \theta\right)\left( \frac{{{\varDelta}} x_{b}}{L}-2\sin \theta\right) $$
(3)

where S = b ⋅ d is the cross sectional area of the guide beam.

3.2 Modeling of folding beam

The deformation of the folding beam is shown in Fig. 4, in which Ls1, Ls3 and Ls5 represent the length of each beam respectively, and Δxf is the displacement of the folding beam due to the external force Fp. The folding beam mechanism can be decomposed of three cantilevers ab, cd and ef, and the deformation of each cantilever is \({{\varDelta }} x_{L_{s1}}\), \({{\varDelta }} x_{L_{s3}}\) and \({{\varDelta }} x_{L_{s5}}\) respectively. According to the superposition principle, the total deformation Δxf of the folding beam mechanism and stiffness kf of mechanism can be derived as following [27]:

$$ \begin{array}{@{}rcl@{}} {{\varDelta}} x_{f}& =& {{\varDelta}} x_{L_{s1}} + {{\varDelta}} x_{L_{s3}} + {{\varDelta}} x_{L_{s5}} \end{array} $$
(4)
$$ \begin{array}{@{}rcl@{}} \frac{1}{k_{f}} &=& \frac{1}{k_{ab}} + \frac{1}{k_{cd}} + \frac{1}{k_{ef}} \end{array} $$
(5)

ab, cd and ef beams can be regarded as semi-fixed beam model under a concentrated load force 2Fp, as shown in Fig. 5. In Fig. 5a, the length of the fixed beam is 2Lsi, and the middle of the beam is subjected to a downward concentrated load 2Fp. Under the force of 2Fp, the maximum deformation of the beam is δmax. If the fixed beam is spilt in the middle, a semi-fixed beam can be obtained as shown in Fig. 5b. The bending moment of the semi-fixed beam at the fixed end is:

$$ M_{o} = \frac{F_{p} L_{si}}{2} $$
(6)
Fig. 4
figure 4

Components of folding beam

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Fig. 5
figure 5

Beam analysis. (a) Fixed-fixed beam under transverse loading. (b) The model of semi-fixed beam

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The shear force is:

$$ R_{s} = F_{p} $$
(7)

The axial force can be neglected because it is too small compared with shear force and bending moment.

In a semi-fixed beam, the maximum displacement δmax is composed of the displacement δM0 caused by the bending moment M0 and the displacement δs caused by the shear force Rs. Therefore, the following equation can be obtained:

$$ \delta_{max} = \delta_{M_{0}} + \delta_{s} $$
(8)

According to Hooke’s Law:

$$ \frac{1}{k_{max}} = \frac{1}{k_{M_{0}}} + \frac{1}{k_{s}} $$
(9)

For a semi-fixed beam, because of bending moment the maximum deflection occurs in the middle of the beam:

$$ \delta_{M_{0}} = \frac{F_{p} L_{si}^{3}}{12EI} $$
(10)

Therefore, the stiffness caused by the bending moment of the fixed beam can be expressed as:

$$ k_{M_{0}} = \frac{12EI}{L_{si}^{3}} $$
(11)

where Lsi is the length of the semi-fixed beam. For semi-fixed beams, the stiffness constant caused by bending moment is half of that of fixed beams. For semi-fixed beams, the maximum deflection due to shear force is:

$$ \delta_{s} = \frac{6F_{s} L_{si}}{5AG} $$
(12)

where G is the shear modulus, G = E/2(1 + μ), μ is the Poisson’s ratio, and A is the cross section area of the beam. According to known conditions, Fs = Fp/2, the results are as follows:

$$ \delta_{s} = \frac{6}{5} \frac{(1+\mu)F_{s} L_{si}}{AE} $$
(13)

The stiffness is:

$$ k_{s} = \frac{5}{6} \frac{AE}{(1+\mu) L_{si}} $$
(14)

By adding the above formula into (5), we can get the following results:

$$ \begin{array}{@{}rcl@{}} \frac{1}{k_{f}} & =& \frac{L_{s1}^{3} + L_{s3}^{3} + L_{s5}^{3}}{12EI} \\ &&+ \frac{6(L_{s1} + L_{s3}+L_{s5}) + 6\mu(L_{s1} + L_{s3}+L_{s5})}{5EA} \end{array} $$
(15)

The relationship between the output force Fp and the input displacement Δxf of the folded beam can be obtained by combining the above formula.

$$ F_{p} = k_{f} {{\varDelta}} x_{f} $$
(16)

When a branch chain of the CFM is subjected to an external force, the deformation of the bistable beam equals to the deformation of the folded beam, namely Δxb = Δxf. By adding the above formula into (3) and (16), the relationship between the force of a branch chain F1 and the input displacement Δx can be obtained:

$$ \begin{array}{@{}rcl@{}} F_{1} &=& F_{n} + F_{p}\\ &=&ES \frac{{{\varDelta}} x}{L}\left( \frac{{{\varDelta}} x}{L} - \sin \theta\right)\left( \frac{{{\varDelta}} x}{L} - 2\sin \theta\right) + \left( \frac{L_{s1}^{3} + L_{s3}^{3} + L_{s5}^{3}}{12EI}\right.\\ &&\left.+ \frac{6(L_{s1} + L_{s3}+L_{s5}) + 6\mu(L_{s1} + L_{s3}+L_{s5})}{5EA}\right)^{-1} {{\varDelta}}_{x} \end{array} $$
(17)

4 FEA verification

In this section, the FEA simulation is utilized to verify the correctness of the theoretical analysis. The adopted material is PLA-ST. The material properties are with Young modulus 1477 MPa, Poisson’s ratio 0.3, and the density 1.25 g/cm3. The original dimensional parameters of the designed CFM are listed in Table 1.

Table 1 Original architecture parameters of the CFM
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The ANSYS Workbench is adopted to verify the performance of bistable and folding beams and the simulation results are depicted in Fig. 6. Referring to the Fig. 6a, the critical buckling force occurs at point of (1.3, 8.46) in theoretical which very closes to the point of (1.2, 9.99) in FEA results. At the same point in folding beam mechanism, the force value 2.86 N is also closed to the 3.09 N. In addition, the slope of both negative stiffness and positive stiffness in theoretical and FEA simulation are very consistent. Referring to Fig. 7, it can be concluded that theoretical and simulation results are both present constant force characteristic and the trends of the two curves are closely within the region of 1.3–5.28 mm. The differences between the theoretical and FEA results are acceptable because of the established mathematic model ignores the effects of intermediate beams. However, the flatness of constant force travel range is poor, which is not suitable for the application requirements. Therefore, it is necessary to optimize architecture parameters to eliminate/reduce the force variation in the specified constant force range.

Fig. 6
figure 6

FEA verification. (a) Theoretical and FEA results comparison of bistable mechanism. (b) Theoretical and FEA results comparison of folding beam mechanism

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Fig. 7
figure 7

Theoretical and FEA results comparison of the designed CFM

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5 Parameter optimization

The performance and dimensional size of the CFM is affected by the designed architecture parameters. For example, the parameter of inclination angle, length, and width of the CFM plays an important influence on the constant force travel range. The design goal of CFM is taken as an end-effector which mounted at the end of robotic arm to implement polishing operation. Hence, the size compactness and mass of the CFM need to taken into consideration in terms of robotic loading capacity. In addition, considering the manufacturing method in this research, the width d of the flexure beam is not less than 1 mm. Because of the structural limitation, the length of L is no longer than 40 mm. The bigger inclination angle of the bistable beam, the larger negative stiffness displacement can be generate, but the larger driving force required to generate buckling. To reduce the manufacturing error, the inclination angle should be larger than 4∘. To eliminate force variation, the aforementioned design objectives and constraint conditions are summarized as follows:

$$ \begin{array}{@{}rcl@{}} Objective: min {\sigma} &=& \sqrt{\frac{1}{N}{\sum\limits_{1}^{N}} (F_{x} - \mu)^{2}} \end{array} $$
(18)
$$ \begin{array}{@{}rcl@{}} \mathit{Subject to}&=& \left \{ \begin{array}{lr} 15\leq L_{s1}\leq 20, \\ 2\leq L_{s3}\leq 5, \\ 15\leq L_{s5}\leq 20, \\ 1\leq b\leq 3, \\ 1\leq d\leq 15, \\ 20\leq L \leq 40, \\ 4^{\circ} \leq \theta \leq 7^{\circ}, \end{array}\right. \end{array} $$
(19)

where σ is the mean square error, Fx represents the force value extracted in a given interval, and μ is the average force value of selected region. When the value of σ is small or becomes zero, it means that force variation is reduced and eliminated.

Taking all the issues into account, PSO method is adopted to find the optimal solution of each parameter. In this research, the 200 sampled points are randomly selected in the range of 2.5 to 4.5 mm. The obtained optimal parameters are listed in Table 2. Substituting the optimal architecture parameters into the analytical model of CFM and conducting the simulation analysis, the FEA results are very consistent with analytical results as shown in Fig. 8. It is further validated the correctness of the theoretical analysis. Referring to the Fig. 8, the line located in the region of 2.5 to 4.5 mm is approximately a horizontal line, in which the maximum value is 12.63 N, the minimum value is 12.43 N and the flatness is 98.41%.

Table 2 Optimized architecture parameters of the CFM
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Fig. 8
figure 8

Comparison of FEA simulation results and optimized results of the CFM

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6 Experimental verification

To test the performance of the proposed CFM, the prototype is fabricated and a series of experimental studies are performed in this section.

6.1 CFM fabrication

To validate the proposed design, the experimental system is built as depicted in Fig. 9, including the designed CFM (fabricated by 3D printing), VCM, grating ruler, force sensor, motor driver, and industrial control computer. To isolation the external disturbances from the environment, the whole experimental system is installed on an anti-vibration optical table. The commercial linear servo amplifier (model: ACJ-055-18) is adopted to drive the VCM (model: TMEC100-015-000) with a nominal travel range of 5 mm. A grating ruler (model: LaRW1-3D, Fagor Automation) is utilized to measure the displacement of the CFM. A force sensor (DS2-XD) with a range of 50 N and a resolution of 0.01 N is chosen to measure the reaction force.

Fig. 9
figure 9

Experimental system

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6.2 Relationship test between force and displacement

During the experimental study, the actuating force and output displacement of the CFM are measured by the force sensor and grating ruler respectively. The comparison of FEA, theoretical calculation and experimental results is shown in Fig. 10. Referring to the experimental results, the constant force range of the CFM starts from 2.68 to 4.81 mm with the constant force is 11.92 N. Compared with the results of finite element analysis and theoretical calculation, the range of constant force increases by 0.18 mm, while the magnitude of constant force decreases by 0.68 N.

Fig. 10
figure 10

Experimental results comparison with FEA simulation and analytical results

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7 Conclusions

This paper presents the design, modeling, simulation, optimization and experimental studies of a spatial CFM for potentially applications in polishing operations. In order to enhance the mechanism stiffness and structure compactness, a parallel spatial CFM based on positive stiffness mechanism and negative stiffness mechanism is designed. To make full use of negative stiffness region, a novel folding beam mechanism with good linearity is designed and the force displacement relationship model referring to the PRB method is established. In addition, to eliminate the force variation, the PSO method is utilized to optimize the architectural parameters. According to the comparison of FEA results of optimized architecture and original design, the flatness in constant travel range is highly improved. Experimental results show that the proposed designed CFM possesses a good characteristic of output constant force. In our future works, multi-body dynamics method will be used to the CFM to better understand its dynamic performance.