Design, Analysis and Experimental Eeri cation of a Epatial Constant Force End-Effector for Polishing/Deburring Operations

Controlling the contact force on workpieces is a challenging task for industrial deburring operations. To solve this issue, a novel constant force mechanism(CFM) based on the combination of positive and negative stiffness mechanism is proposed by using folding beam and bi-stable beam. Without using any additional sensors and control algorithms, the proposed mechanism can produce a travel range in constant force manner. In this paper, the design concepts, analytical modeling, finite element analysis (FEA) simulation and experimental studies are presented and discussed. Firstly, a novel spatial CFM is proposed and using the pseudo rigid body(PRB) to established the mathematical model of the whole mechanism. Then, the FEA simulation is performed to validate the correctness of theoretical analysis. In addition, to eliminate the force variation, particle swarm optimization (PSO) method is utilized to find optimal architectural parameters solutions of the CFM. Finally, the experimental tests are performed to verify the performance of the designed CFM. The configuration design and parameter optimization can be further applied to the design of other CFM mechanisms for polishing operations.


Introduction
Robots are increasingly used for workpieces deburring and polishing to overcome the problems of low production efficiency, high cost and poor consistency in manual polishing[1]- [3]. As pointed in previous research, the contact force between the grinding tool and work-piece is the key factor that affects the polishing quality [4,5]. Excessive or insufficient contact force would greatly impair the quality of the surface roughness. Thus, to maintain the constant contact force in the grinding process to ensure the polishing quality and consistency becomes an urgent issue [6]- [8].
Generally speaking, the constant force output can be realized by two methods [9]. 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 [10,11]. Masoud et al proposed a resultant force control strategy for light abrasive surface grinding. The performance of resultant force control strategy with real-time TDC algorithm is validated by the grinding experimental studies [12]. 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±0.1mm, and the polishing quality of the workpiece surface can be significantly improved [13]. 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 [14]. 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 in robot grinding process [15]. 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 will degrade the force control performance through the force-position hybrid controller to realize active compliance force control [16].
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 [17,18]. 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 [19]. 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 [20]. Liu proposed a new polishing mechanism, and detailed modeling and analysis of hardware circuit structure and force control algorithm design,The polishing actuator has good compliance performance [21]. 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 [22]. 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 [23]. 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.

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 constant force output, the combination of positive stiffness mechanism and 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 travel range constant force output. The force-displacement relationship comparison of designed positive stiffness mechanism and traditional mechanism is depicted in Fig.1 which shows that the folded beam mechanism has a good linearity in long range. The autonomous polishing scenario using CFM system is shown in Fig.2, the end-effector including voice coil motor, CFM and grinding head are installed in the end of robot arm through the connecting flange. When the voice coil motor works, it drives the grinding head to approach the workpiece. As an intermediate part, the CFM placed between the motor unit and grinding tool and the grinding tool mounted on the CFM. Due to the constant force output characteristics of the CFM, the grinding head can maintain a constant force contact with the workpiece in a certain range of motion, so as to finish the precision grinding and polishing.

Analytical design model
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.

Modeling of bi-stable beam mechanism
As shown in Fig.3, buckling and deformation of the bistable beam is resulted by the loading force F n . 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 ∆x b is the displacement of the bistable beam under the force of F n along Y direction.
When the force F n loading on the bistable beam exceeds the critical force of the material, referring to the previous studies the buckling force of the i critical point of the guide beam can be written as [24,25]: where λ n = π, 4.493, 2π, ...,n = 1, 2, 3, ... The negative stiffness of bistable beam k 1 can be expressed as follow: Where E represents the elastic modulus of the material, I = bd 3 /12 is the section moment of inertia of the guide beam, d is the thickness of the beam, b is the width of the beam. Referring to equations (1) and (2), the force-displacement relationship of the bistable beam can be written as: Where S = b · d is the cross sectional area of the guide beam.

Modeling of folding beam mechanism
The deformation of the folded beam is shown in Fig.  4, in which L s1 , L s3 and L s5 represent the length of each beam respectively, and ∆x f is the displacement of the folded beam due to the external force F p . The folding beam mechanism can be decomposed of three cantilevers ab, cd and ef , and the deformation of each cantilever is ∆x Ls1 , ∆x Ls3 and ∆x Ls5 respectively. According to the superposition principle, the total deformation ∆x f of the folding beam mechanism and stiffness k 2 of mechanism can be derived as following [26]: ab, cd and ef beams can be regarded as semi-fixed beam model under concentrated load 2F p , as shown in  In Fig.5(a), the length of the fixed beam is 2L si , and the middle of the beam is subjected to a downward concentrated load 2F p . Under the force of 2F p , the maximum deformation of the beam is δ max . If the fixed beam is spilt in the middle, a semi-fixed beam can be obtained as shown in Fig.5(b). The bending moment of the semi-fixed beam at the fixed end is: The shear force is: 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 δ max is composed of the displacement δ M 0 caused by the bending moment M 0 and the displacement δ s caused by the shear force R s .
According to Hooke's Law: For a semi-fixed beam, because of bending moment the maximum deflection occurs in the middle of the beam: Therefore, the stiffness constant caused by the bending moment of the fixed beam can be expressed as: Where L si 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: where G is the shear modulus, G = E/2(1 + µ), µ is the Poisson's ratio, and A is the area of the beam cross section. According to known conditions, F s = F p /2 , the results are as follows: The stiffness constant is: By adding the above formula into (5), we can get the following results: 12EI + 6(L s1 + L s3 + L s5 ) + 6µ(L s1 + L s3 + L s5 ) 5EA The relationship between the output force F p and the input displacement ∆x f of the folded beam can be obtained by combining the above formula.
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 ∆x b = ∆x f . By adding the above formula into (3) and (16), the relationship between the force of a branch chain F 1 and the input displacement ∆x can be obtained:

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 1477M P a, Poisson's ratio 0.3, and the density 1.25g/cm 3 . The original dimensional parameters of the designed CFM is listed in Table 1. The ANSYS Workbench is adopted to verify the performance of bistable and folding beams and results are depicted in Fig.6. Referring to the Fig.6(a), the critical buckling force point occurs at (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 folded beam mechanism, the force value 2.86N is also closed to the 3.09N . 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 [1.2-5.2] mm. The differences between the theoretical and FEA results are acceptable because of the established mathematic model ignores the effects of intermediate beams. In addition, 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.

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 travel range of constant force. 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 1mm. Because of the structural limitation, the length of L is no longer than 40mm. The bigger inclination angle of the bistable beam, the larger negative stiffness displacement can 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: where σ is the mean square error, F x 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 2.5mm to 4.5mm. The obtained optimal parameters are listed in Tab.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 2.5mm to 4.5mm is approximately a horizontal line, in which the maximum value is 12.63N , the minimum value is 12.43N and the flatness is 98.41%.

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.

Relationship test between force and displacement
During the experimental study, the actuating force and ouput 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 2.68mm to 4.81mm with the constant force is 11.92N . Compared with the results of finite element analysis and theoretical calculation, the range of constant force increases by 0.18mm, while the size of constant force decreases by 0.68N .

Conclusions
This paper presents the design, modeling, simulation, optimization and experimental studies of a spa- tial CFM for potentially application 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 folded 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   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 CFM possesses a good characteristic of output constant force.

Fundings
This work is supported by Huxiang High Level Tal

Conflicts of Interest
The authors declare no conflict of interest.

Author contributions
Bingxiao Ding provided the concept of the work, built up the model, Jiyu Zhao designed mechanism and conducted experimental verification, and Yangmin Li carried out supervision and revision of this paper.

Figure 3
Buckling deformation of bistable beam mechanism.

Figure 4
Resolved components of folded beam.     Structure design of grinding mechanism with constant output force.

Figure 10
Experimental results comparison with FEA simulation and analytical results.