Evaluation on Configuration Stiffness of Overconstrained 2R1T Parallel Mechanisms

Currently, two rotations and one translation (2R1T) three-degree-of-freedom (DOF) parallel mechanisms (PMs) are widely applied in five-DOF hybrid machining robots. However, there is a lack of an effective method to evaluate the configuration stiffness of mechanisms during the mechanism design stage. It is a challenge to select appropriate 2R1T PMs with excellent stiffness performance during the design stage. Considering the operational status of 2R1T PMs, the bending and torsional stiffness are considered as indices to evaluate PMs’ configuration stiffness. Subse-quently, a specific method is proposed to calculate these stiffness indices. Initially, the various types of structural and driving stiffness for each branch are assessed and their specific values defined. Subsequently, a rigid-flexible coupled force model for the over-constrained 2R1T PM is established, and the proposed evaluation method is used to analyze the configuration stiffness of the five 2R1T PMs in the entire workspace. Finally, the driving force and constraint force of each branch in the whole working space are calculated to further elucidate the stiffness evaluating results by using the proposed method above. The obtained results demonstrate that the bending and torsional stiffness of the 2RPU/UPR/RPR mechanism along the x and y -directions are larger than the other four mechanisms


Introduction
Studies show that numerous parallel mechanisms (PMs) for diverse applications have been proposed so far.Among different types of lower-mobility PMs, the tworotational-degrees-of-freedom and one-translationaldegree-of-freedom (2R1T) PM are the most important ones.It should be indicated that 3RPS PM [1] is a typical 2R1T-PM.In 1983, Hunt [1] proposed the 3RPS PM that widely attracted many researchers.In this regard, investigations have been carried out to study 2R1T PM from different aspects, including synthesis, kinematics, parasitic motion, singularity analysis, and dimension synthesis [2][3][4][5][6][7][8][9][10].Recently, modified 2R1T PMs have been proposed, which have been applied widely in diverse fields such as A3 machine tool head based on the 3RPS PM [2,4,11], Sprint Z3 tool head based on the 3PRS PM [12,13], Tricept [14], TriVariant [15], five-DOF parallelserial manipulators based on 3UPS-UP and 2UPS-UP PMs, and Exechon five-axis machining center based on the 2UPR-SPR PM [16,17], which are typical applications of the 2R1T-PMs.It is worth noting that R, P, S, and U denote revolute, prismatic, spherical, and universal joints, respectively.
The configuration stiffness of a mechanism refers to the end-effector displacement caused by the deformation of its parts under the action of external forces, where the structure shape and size are not determined yet, which plays a decisive role in the motion accuracy of the mechanism.However, during the configuration design of the parallel robot mechanism, no effective method has yet been proposed to evaluate the configuration stiffness of the robot.
Several approaches can be applied to calculate the stiffness matrix.These approaches mainly differ in the model assumptions and computational techniques [33][34][35][36].
Reviewing the literature indicates that the finite element method (FEM) [37,38], matrix structure method (MSM) [39], virtual joints method (JVM) [40][41][42], and screwbased method (SBM) [43][44][45][46][47] are the most commonly used approaches in this regard.Generally, the finite element method is applied at the final design stage for the verification and component dimensioning.Moreover, the matrix structure method needs tedious calculations.The JVM and SBM methods have a clear physical interpretation and can accurately reflect the correlation between the stiffness of PMs and the stiffness along the axes of the constraint wrenches.In the latter two methods, actual stiffness models of actuation units, transmissions, passive joints, and connecting links are considered to establish the overall stiffness matrix of the robot [48,49].The singular value decomposition can be carried out on the stiffness matrix to get the direction of the maximum and the minimum stiffness and obtain the maximum and minimum deformations accordingly [50][51][52].However, the stiffness matrix model involves the robot parts' specific structural shape and size, while the modeling process is quite complicated.Meanwhile, this method does not guarantee the same size for different mechanisms.Since the stiffness matrix has different dimensions, it is impossible to judge the maximum and minimum stiffness accurately.Hu and Huang [53] obtained the stiffness model of a 2RPU-UPR (U = universal joint, P = prismatic joint, R = revolute joint) over-constrained parallel manipulators.Yang and Li [54] propose a unified method for the elastostatic stiffness modeling of over-constrained parallel manipulators.Ding et al. [55] analyzed the accuracy of an over-constrained Stewart platform with actuation stiffness.Cao et al. [56,57] obtained the stiffness model of over-constrained parallel manipulators by using an energy method with less than 3% accuracy loss only under an external wrench.Cao et al. [58] extended the approach by considering the weights of the links in which the weight of each limb is distributed between the mobile platform frame and the base.Finally, Klimchik et al. [59] derived the stiffness model of NAVARO II, which is a novel variable actuation mechanism based on active and passive pantographs.The connection method of multiple closed-loop chains is illustrated for an external wrench on the mobile platform via SMA; however, the dimensions of the matrices to be computed were relatively high.The fundamentals of the SMA methodology are given in Ref. [60].
A variety of stiffness evaluation indexes of parallel mechanisms have been proposed, such as eigenvalue index [61,62], determinant value index [63], trace of matrix [63], Weighted index of trace [64], main diagonal index of stiffness matrix [65], virtual work index [61].However, the above stiffness indexes are obtained based on the overall stiffness matrix of the mechanism, these indexes do not consider the influence of structural composition on the mechanism stiffness in all directions.And this cannot reflect the weaker direction of the PM stiffness in the mechanism design stage.From the structural composition point of view, it can be concluded that the PM has weak stiffness in bending and torsional direction, while the tensile and compressive stiffness is relatively strong, because the axial load is generally borne by driving forces of all branches together, but it can be seen as a cantilever system when the bending forces and torsional torques are imposed.For this reason, this study proposes bending and torsional stiffness indexes of parallel mechanisms considering the structural constraints, and compares the bending and torsional stiffness of several 2R1T parallel mechanisms to select the preferred one with good stiffness performance.
As the extension length of the working arm becomes longer, the bending and torsional stiffness gradually decrease, and the imposed load on the branch and part of the branch are entirely different.This phenomenon mainly originates from different compositions and distributions of the constraint forces/couples under the action of the external load at the end.For example, for the 2UPR-SPR PM [66] that constitutes the Exechon fiveaxis hybrid machining center, all three driving branches bear the external load of the constrained space and the driving space simultaneously.However, for the 3UPS-UP PM [67] that constitutes the Tricept five-axis hybrid robot, the constraint branch UP completely tolerates the external load in the constrained space.Meanwhile, different robots may have different structural branches, actuation unit stiffness, transmission devices, and connecting links.This issue is significantly more pronounced for the stiffness of the driving unit and transmission device along the non-main direction so that the stiffness performance of different mechanisms may be different.Therefore, evaluating the configuration stiffness considering the structural constraints of the parallel mechanism, lacks simple evaluation criteria.Configuration stiffness means the mechanism stiffness during the design stage where the structure shape and size are not determined yet.Accordingly, it is an enormous challenge to select the best mechanism with optimal stiffness performance during the design stage.
In the present study, the PMs' bending and torsional stiffness indices will be defined.Then these indices will be applied to a five-axis hybrid robot in an engineering case study.For the studied case, values of the driving and structural stiffness of each part in the branch that affect the overall stiffness of the mechanism will be directly defined.Then the global stiffness of the parallel mechanism can be obtained, and all the bending deformation and torsional deformation are obtained based on the over-constrained PM's force analysis.What's next, all the driving and constraint forces/couples are obtained, which are used to explain the reasons for the configuration stiffness evaluating results.Finally, through Ansys software, the stiffness of five parallel mechanisms is simulated to verify the correctness of the stiffness evaluating indices and methods.This article is expected to provide a new and simple way to evaluate the stiffness performance of PMs during the mechanism design stage.

Stiffness Evaluation Index and Evaluation Criterion
In the present study, the bending stiffness of PMs is defined as the capability of the mechanism to resist the bending deformation, specifically under the action of a lateral unit force on the moving platform.Under the action of a unit lateral force on the moving platform, the linear deformation of the parallel mechanism based on the global stiffness model is solved according to the defined branch stiffness.The average linear deformations of the mechanism indicate values of the bending stiffness of PMs.The bending stiffness of the mechanism can be divided into the stiffness in the lateral x and y-axis, respectively.Figure 1 shows that at the origin of the moving coordinate system at the end of the mechanism, the lateral unit force is exerted along the x and y-direction, respectively.Furthermore, torsional stiffness is defined as the capability of the mechanism to resist torsional deformation, specifically under the action of a unit torque along the normal line of the moving platform on the moving platform.Under the action of a unit torque along the moving platform's normal direction, the parallel mechanism's angular deformation based on the global stiffness model is solved according to the defined branch stiffness.Similarly, the average of angular deformations of the parallel mechanism can be calculated, indicating the values of PMs' torsional stiffness.
Based on the above definition, the main factors affecting the bending and torsional stiffness of the 2R1T PM are the stiffness of each branch.Considering the actual PM-based designed equipment, the branch stiffness is affected by the structural stiffness of the links, joints, and the stiffness of driving units.And based on the engineering experience, considering the strength and weakness of the stiffness of these branch parts, different values of the stiffness are defined to get the linear and angular deformations of the mechanism so that the PM with relatively better stiffness performance can be selected.For example, the bending and torsional stiffness of the linear driving unit is relatively weak among those structures mentioned above in each branch, so their defined values should be smaller.
In order to compare the stiffness performance of the PMs qualitatively, the global bending and torsional performance indices can be mathematically expressed as follows: where Γ ω and Γ φ are the bending and torsional stiffness performance indices, respectively.Meanwhile, Γ ω can be divided into Γ ω x and Γ ω y along in x-and y-directions, respectively.Moreover, ω i and φ i denote linear deformation and torsional angle deformation values at the sample point i, respectively.i is the number of samples, and w denotes the workspace.
(1)  Suppose that each branch's upper and lower connecting links are cylindrical rods with structural parameters given in Table 1, where μ is Poisson's ratio, 2a, and 2b are the length of the triangular base of the moving and fixed platform, respectively.Moreover, φ is the angle between two hypotenuses of a fixed platform, h is the initial distance between moving and fixed platforms, l i1 (i=1,2,3,4) is the initial length of the lower connecting link of each branch, d i1 and d i2 are the cross-section diameters of the upper and lower connecting links, respectively.And the diameter of P joint is d p , length is l p .
Let α and β be angled rotating around the axes of two rotational DOFs r 1 and r 2 , with initial values of α=β=0 °.Then the length of each branch (i.e., l 1 , l 2 , l 3 , and l 4 ) can be determined through the inverse position solution of the parallel mechanism.

Configuration Stiffness Analysis
The five 2R1T PMs have common features, including linear drive unit, prismatic joint, revolute joint and rod.Based on the experience of structural design of the robots, for example, the flexural stiffness of the driving structure should be lower, since the transmission device has weak stiffness in the lateral direction.Consequently, simplified models of five PMs with the same size are designed, based on which the stiffness values of each part are defined.Table 2 gives the specific values of four kinds of branch stiffness affecting the bending and torsional stiffness of the mechanisms.
The given stiffness of the rod includes axial stiffness, flexural stiffness and torsional stiffness.Moreover, structural stiffness of the R joint includes torsional stiffness, tension and compression stiffness.Finally, the structural stiffness of the P joint includes flexural stiffness and torsional stiffness.The axial stiffness of the P joint refers to the capability of the driving device, which is along the axial direction of driving force.
The compliance matrices of revolute joints, prismatic, and each branch's lower rod are constant, while the compliance matrix of the branch's upper rod is variable, which is related to the change of the branch length.Thus, a coefficient λ i is introduced, then the i-th (i=1,2,3,4) branch's upper rod compliance matrix C is of the mechanism can be written as: where λ i is the length ratio, λ i = l is / l is0 .And l is0 is the upper rod's length of the i-th branch, l is is the initial length of the upper rod in the i-th branch, k f is the bending stiffness of the upper rod; k a is the axial stiffness of the upper rod; k t is the torsional stiffness of the rod.
Taking the 2RPU/UPR PM as an example, the moving platform's linear deformation and torsional deformation under the action of external force will be solved below.
The relationship between the deformation and external force of the i-th branch's end is: where, F i is the applied force at the end of the i-th branch, X i is the total deformation at the end of the i-th branch, C i is the whole compliance matrix of the i-th branch.
The relationship between the force F ij applied at the point a ij and the applied force F i at the point a i in the i-th branch shown in Figure 3 can be expressed as (3) where j J i is the force transformation matrix between F i and F ij , which can be expressed as j i R represents the orientation matrix of the frame a i - u i v i w i attached at point a i with respect to frame a ij -u ij v i- j w ij attached at point a ij , j i P denotes the position vector from point a ij to a i in the frame a ij -u ij v ij w ij , S( j i P) is the anti-symmetric matrix composed of vector j i P. The deformation X ij at point a ij can be calculated under the action of force F ij applied at the joint j: Substituting Eqs. ( 5) ~ (7) into Eq.( 4), the deformation of the i-th branch's end can be formulated as: (5) Supposed that C x represents the compliance matrix of branch's lower rod, C R represents the compliance matrix of R joint, C P represents the compliance matrix of prismatic joint.Based on Eq. ( 8), the deformation at the end of the R 1 P 1 U 1 branch can be calculated as where Then the deformation of the branch's end along the direction of the constraint force can be expressed as: where G 1 f is the unit screw system expressed in the frame At the same time, the force F 1 can be expressed as: Substituting Eq. ( 11) into Eq.( 10), X 1  f can be expressed as: Then the stiffness of the R 1 P 1 U 1 branch can be obtained: In Refs.[71,72], the over-constraint wrench system and the spatial composite elastic deformation of the branch members of the parallel mechanism were considered, the whole stiffness matrix of the PMs was (8) .  where G F f is the mapping matrix of the helical magnitude f of each constraint force of the parallel mechanism and the external force F of the parallel mechanism; K i is the stiffness matrix of the i-th branch of the parallel mechanism.
Under the action of external force F, the deformation D of the moving platform and the external force F have the following relationship: For five different mechanisms, variation range of the angles α and β are confined to [− π/6~π/6] and [− π/9~π/9], respectively, which can be obtained based on the workspace of the PMs.At the same time, the parameter h of the parallel mechanism is set to be 0.335 m.Using the principle of the "layered slice", the linear deformation distribution of the moving platform of the mechanism can be obtained combined with the inverse position solution.
Under the action of an external load with 10 N exerted on the origin of the moving coordinate system along the x-direction, each PM's linear deformation in the x-direction is calculated based on Eq. ( 1), as shown in Figure 4.
Similarly, under the action of an external load with 10 N exerted on the origin of the moving coordinate system along the y-direction, each PM's linear deformation in the y-direction is calculated based on Eq. ( 1), as shown in Figure 5.
Under the action of an external load with 10 N•m exerted on the origin of the moving coordinate system around the z-direction, each mechanism's angular deformation in the z-direction is calculated based on Eq. ( 2), as shown in Figure 6. ( 14)  In what follows, the influence factors of the configuration stiffness will be analysed from the point of constraint forces and deformations of the branches, thus the correctness of the above conclusions can be verified.

Force Analysis of the Branches
Considering redundant and typical constraints, the force analysis of the over-constrained PMs is a statically indeterminate problem with complicated calculations.Accordingly, an analysis method based on ADAMS simulation software is applied in the present study.In this regard, a method [47] has been proposed to establish the rigid-flexible coupling simulation model of over-constrained PMs.However, the simulation model can only stay in a single configuration when PMs contain translational joints.In other words, this method cannot realize the force simulation in the whole workspace.In order to resolve this shortcoming, an improved method is proposed to establish the rigid-flexible simulation model of over-constrained PMs and realize the continuous force simulation analysis of the PMs in the whole workspace based on the method proposed in Ref. [47].
Accordingly, the rigid-flexible coupling model of the over-constrained mechanism has been established in ADAMS software.It is assumed that the moving platform and the fixed platform are rigid bodies, while the connecting links are flexible bodies.For the revolute joints, including the universal joints and the spherical joints, since the position of the rotating axis does not change relative to the rotating links, the position of the interface node within the modal center file relative to the link is fixed.Consequently, only one interface node should be set to connect the position of each revolute joint and simulate the revolute contact.However, for prismatic joints such as the guide rail and the slider, when the slider translates relative to the guide rail, the contact position between the guide rail and the slider constantly changes during the movement.In ADAMS software, since only a single position can be chosen when establishing the kinematic joints in the flexible body, only the force performance of prismatic joints in the current state can be simulated, and the entire sliding process cannot be simulated.If multiple interface nodes are set up for the prismatic joints in the modal neutral file, P joints can be established for the slider and the guide rail at each node position.In the simulation process, when the slider moves to a specific position relative to the guide rail, the prismatic joints at the corresponding node position are set to be activated, and the other joints are deactivated.Similarly, when the slider moves to the next position, the prismatic joints in the previous position become invalid, while the prismatic joints in the current position become effective.The whole process can be considered the discretized contact position of the flexible body into multiple interface nodes.The greater the discretization degree, the more accurate the simulation results, but the greater the corresponding computational costs.Multiple interface nodes the discretization degree.
Figure 7 illustrates the established flexible model of the slider containing multiple interface nodes.The calculation process is as follows.
At the first step, the multiple interface nodes were introduced to ADAMS software, and then the joints, driver, and external force were applied to the interface nodes of the slider.Meanwhile, the prismatic joints and drivers were exerted to middle multiple interface nodes, which show in Figure 8.
The calculations assume that the slider is a cylindrical rod with a diameter D and length L.Moreover, the translational speed, motion time, and the applied force to the end of the slider are v, t, and F, respectively.The slider can be simplified to a cantilever beam, and the end deformation along the force axis F can be calculated.When the slider moves to the left, the end deformation gradually decreases because the cantilever's length gradually decreases during the movement.
Deformation of the cantilever end can be calculated through the following Eq.( 16): where I=πD 4 /64 and E are the moment of inertia of the slider and modulus of elasticity, respectively.Table 3 is the flexible model of the P joint's settings of parameters.
Deformation of the cantilever end can then be obtained either from simulations or Eq. ( 16). Figure 9 presents the obtained results accordingly.
Figure 9 reveals that the theoretical and simulation values are consistent, the maximum error between them is less than 6%, resulting from the discretization degree of interface nodes and finite element size, verifying the accuracy of the established model for the prismatic joint in ADAMS software.
Based on the modeling mentioned above method, the rigid-flexible coupling model of the five PMs is established.Figure 10(a)-(e) shows the established models for five types of 2R1T PMs, which can realize the force simulation analysis in the continuous motion process.
In the moving coordinate system o-xyz, the load $ F = (10 N 10 N 10 N; 10 N • m 10 N • m 10 N • m) is applied on the origin of the moving platform for the 2RPU/UPR mechanism.The following motion trajectory is considered in the simulations for the moving platform: where t=0~1 s is the motion time.
Figure 11(a)-(c) shows variations of the magnitude of driving and constraint force/couple obtained from the rigid-flexible coupling simulations.
In Fig. 11, f a,i represents the magnitude of the driving force of the i-th branch, f r,i the magnitude of the constraint force of the i-th branch, m r,i the magnitudes of the constraint couple of the i-th branch for the 2RPU/UPR mechanism.
In order to verify the analysis accuracy, the weighted generalized inverse method [71,72] is used to analyse the theoretical force of the typical over-constrained PMs.Equation ( 18) can be expressed as follows.
where G F f is the mapping matrix from f to $ F , $ F is the external force/torque applied on the moving platform, f is the column vector composed of the magnitudes of the driving and constraint force/couple, and W is the weighting matrix determined by the structural stiffness of each branch.(16) w B = (FL 3 )/(3EI), (17)    P(x) = 0, P(y) = 0, P(z) = 0.335 + 0.005t, (18)  When the mechanism is in the initial state (i.e., t= 0 s and α=β=0°), magnitudes of the driving force and constraint force/a couple of each branch obtained from simulations and theoretical calculations are listed in Table 4.
It is observed that the error between the theoretical and simulation value is tiny, demonstrating the accuracy of the theoretical and simulation analysis.
Similarly, the accuracy of the other four mechanisms can be verified.

Comparison of the Branch's Constraint Force/Couple
Using the principle of the "layered slice", the average amplitude of each branch's driving and constraint force/ couple of the parallel mechanism at a certain height is obtained combined with the inverse position solution based on the abovementioned method.
Table 5 gives the average magnitude of each branch's the driving and constraint forces/couples of each parallel mechanism in the entire workspace when h=0.335 m, In Table 6, 2UPU/SP (2) means that the mechanism is at the position where the x and X-axis intersect, $ r 12 and $ r 22 are 0; 2UPU/SP (1) means that the mechanism is at the position where the x and X-axis are parallel, $ r 11 and $ r 21 are 0. The position of the 2UPU/SP (1) occupies a small part of the workspace under the designation, so the PM's branch deformation is mainly determined by the deformation at the 2UPU/SP (2) position.
Table 6 gives the average magnitude of each branch's the driving and constraint forces/couples of each parallel mechanism in the entire workspace when h=0.335 m under the action of a lateral force $ Fy = (0 10 N 0; 0 0 0) exerted on the moving platform's origin.
Table 7 gives the average magnitude of each branch's the driving and constraint forces/couples of each parallel mechanism in the entire workspace when h=0.335 m under the action of a lateral force $ Fz = (0 0 0; 0 0 10 N•mm) exerted on the moving platform's origin.
By observing Tables 5, 6, 7, we can get the magnitudes of the driving and constraint forces/couples on the branches of the five PMs under the action of same external force.By comparing the magnitudes of the driving and constraint forces/couples on the same branch, the main factors affecting the deformation of the branch can be obtained.

Comparative analysis of the driving and constraint
forces/couples of branches under the action of external force $ Fx .
The reason of this result would be analyzed from the point of constraint force/couple supplied by the branches, since the axial stiffness of the limb is relatively large.
It can be seen from Table 5 that when the same external force $ Fx is applied on the moving platform origin, the    and $ r 32 .Among them, the constraint forces against this external force for the 2RPU/UPR/RPR PM are obviously smaller than that of other parallel mechanisms.Therefore, among the five parallel mechanisms, 2RPU/UPR/ RPR has the smallest deformation and the largest stiffness in the X-direction.

Comparative analysis of the driving and constraint
forces/couples of branches under the action of external force $ Fy .
It can be seen from Table 6 that when the same external force $ Fy is applied, the constraint forces that mainly resist this external force for the 2RPU/UPR PM are $ r 11 and $ r 21 , the constraint forces that mainly resist this external force for the 2RPU/UPR/RPR PM are $ r 11 ,$ r 21 and $ r 42 , and the constraint force that mainly resist this external force for the 3UPS/UP PM is $ r 42 , the constraint forces that mainly resist this external force for the 2UPR/SPR PM are $ r 11 and $ r 21 , and the constraint forces that mainly resist this external force for the 2UPU/SP PM are $ r 11 , $ r 21 and $ r 31 .Among them, the constraint forces against this external force for the 2RPU/UPR/RPR PM are obviously smaller than that of other parallel mechanisms.Therefore, among the five parallel mechanisms, 2RPU/UPR/ RPR has the smallest deformation and the largest stiffness in the Y-direction.

Comparative analysis of the driving and constraint
forces/couples of branches under the action of external force $ Fz .
It can be seen from Table 7 that when the same external force $ Fz is applied, the constraint forces that mainly resist this external force for the 2RPU/UPR PM are $ r 12 , $ r 22 , $ r 31 and $ r 32 , the constraint forces that mainly resist this external force for the 2RPU/UPR/RPR PM are $ r 12 , $ r 22 , $ r 31 , $ r 32 and $ r 43 , and the constraint force that mainly resist this external force for the 3UPS/UP PM is $ r 43 , the constraint forces that mainly resist this external force for the 2UPR/SPR PM are $ r 11 , $ r 12 , $ r 12 , $ r 22 and $ r 31 , and the constraint forces that mainly resist this external force for the 2UPU/SP PM are $ r 11 , $ r 12 , $ r 21 , $ r 22 , $ r 31 and $ r 32 .When an external couple is applied to parallel mechanisms, its deformation is complicated, which is the result of the joint action of constraint forces and couples.Therefore, although the constraint couples of 2UPU/SP PM are small, its constraint forces cause large torsional deformation, so its overall deformation is large.For the 3UPS/UP PM, though the value of constraint couple is the largest, there are fewer passive joints within the UP limb, so the torsional deformation caused by the constraint couple is not the largest.
For the 2RPU/UPR, 2RPU/UPR/RPR and 2UPR/SPR PMs, the constraint forces of 2RPU/UPR/RPR PM against this external force are obviously less than that of the other two parallel mechanisms.Therefore, among the five parallel mechanisms, 2RPU/UPR/RPR has the smallest deformation and the largest stiffness in the Z-direction.
Therefore, the mechanism 2RPU/UPR/RPR can preferably be used as the PM part of the five-DOF hybrid machining robot, which is consistent with the analysis results in Section 4.

Effect of Component Stiffness on Overall Stiffness
Taking 2RPU/UPR parallel mechanism as an example, using the control variable method, only the stiffness of a certain component is changed, and the deformation changes of the parallel mechanism in the X, Y, Z directions are observed under the action of a same force.
External force $ F = (10 N 10 N 10 N; 10 N •m 10 N •m 10N •m) exerted on the moving platform origin.When h=0.335 m, the average deformation of the parallel mechanism in the X-direction in the entire workspace is shown in the following Figure 12.
In Fig. 12, k la stands for axial stiffness of rod, k lf stands for flexural stiffness of rod, k lt stands for torsional stiffness of rod, k da stands for axial stiffness of drive structure, k df stands for flexural stiffness of drive structure, k dt stands for torsional stiffness of drive structure, k rc stands for tension and compression stiffness of R joint, k rt stands for torsional stiffness of R joint, k pf stands for flexural stiffness of P joint, k pt stands for torsional stiffness of P joint.And stiffness variation coefficient denotes each structure stiffness change multiples, which is a dimensionless value.
Under the action of same external force, when h=0.335 m, the average deformation of the parallel mechanism in the Y-direction in the whole workspace is shown in Figure 13.
Under the action of same external force, when h=0.335 m, the average deformation of the parallel mechanism in Z-direction in the whole workspace is as shown in Figure 14.
It can be seen from the observation that the X-direction deformation of the parallel mechanism is more sensitive to the numerical changes of k la , k da and k rc .The larger the values of k la , k da and k rc are, the smaller the X-direction deformation of the parallel mechanism is.The Y-direction deformation of the parallel mechanism is sensitive to the numerical changes of k lf and k df .The larger the values of k lf and k df are, the smaller the Y-direction deformation of the parallel mechanism is.The Z-direction deformation of the parallel mechanism is sensitive to the numerical changes of k la , k da , k df and k rc .The larger the values of k la , k da , k df , and k rc are, the smaller the Z-direction deformation of the parallel mechanism is.At the same time, it can be seen from observation that not all stiffness values are greater, the overall stiffness of the parallel mechanism is greater.
Based on the existing stiffness, if the stiffness of the parallel mechanism needs to be improved, the values of k la , k da , k df , and k rc should be increased.Therefore, when designing the structure of parallel mechanism, increasing the section size of the branches, the driving structures and the revolute points can be considered, or

Simulation Verification
In order to verify the correctness and rationality of the proposed stiffness performance indices, considering the five-DOF hybrid processing robot as the application target, simple engineering three-dimensional models of five 2R1T PMs, including 2RPU/UPR, 2RPU/UPR/ RPR, 3UPS/SP, 2UPR/SPR, and 2UPU/SP, are established, where the structure of branch screws and screw nuts, branch guide rails and sliding blocks, and revolute joints connecting each branch to the moving and fixed platforms are taken into account.Five simulation calculation models are established by using the Ansys software.It should be indicated that kinematic joints are added, the sliders and the guide rails are locked, and friction is added at each revolute joint.The forces 100 N, 100 N, and a couple of 100 N•m are exerted along the coordinate system's three coordinate axes x, y, and z on the moving platform.The stiffness simulation of the   In Figure 20, d and β are kept at a constant, that is, d=290 mm and β=0.In Figure 21, d and α are kept at a constant, that is, d=290 mm and α=0.In Figure 22, α and β are kept at a constant, that is, α=0 and β=0.
It can be seen from Figures 20, 21, 22 that the 2RPR/ UPR/RPR mechanism has better bending stiffness in the x and y-directions and torsional stiffness in the z-direction than the other four PMs, consistent with the theoretical analysis result.Therefore, the stiffness evaluation indices and methods are rational.

Conclusions
To address the challenge of evaluating the configuration stiffness performance of PMs, new evaluation indices and a corresponding method are proposed.Utilizing these, the stiffness performances of five PMs are compared.
(1) Bending and torsional stiffness evaluation indices for PMs have been proposed, along with a method for assessing the PMs' configuration stiffness.Furthermore, the established model is applied to a real case study.Then various driving and structural The proposed stiffness evaluation method can also be applied to other PMs.In our future work, we will evaluate more PMs and select those with greater configuration stiffness.

Figure 1
Figure 1 Schematic diagram of the parallel mechanism

Figure 2
Figure 2 Schematic configurations of five 2R1T PMs

Figure 3 Figure 4 Figure 5 Figure 6 ( 1 )
Figure 3The coordinates in the i-th branch

Figure 7 Figure 8
Figure 7 Flexible model of the slider containing multiple interface nodes

Figure 10
Figure 10 Rigid-flexible coupling models of five types of 2R1T PMs

Figure 11
Figure 11 Variations of the magnitudes of the driving and constraint forces/couples

Figure 13 Figure 14
Figure13 The deformation changes in Y-direction of 2RPU/UPR PM when the stiffness value changes

Figure 15 Figure 16
Figure 15 Deformation of the 2RPU/UPR PM in different directions , 17, 18, 19 show the obtained results.To further verify the stiffness analysis results, the stiffness simulation analyses in different configurations are also carried out in the whole workspace.And the maximum deformation in x, y and z-axis, with respect to the variable α, β, and d, are shown in Figures 20, 21 , 22, respectively.Where α, β, and d represent rotation angle around the x-axis, rotation angle around the y-axis, and distance from point o to O, respectively.

Figure 17 Figure 18
Figure 17 Deformation of the 3UPS/UP PM in different directions

Figure 19 Figure 20 Figure 21
Figure 19 Deformation of the 2UPU/SP PM in different directions

Table 1
Structure parameters of parallel mechanisms

Table 2
Definition of four types of branch stiffness

Table 3
Settings of parameters of the flexible model of the P joint Figure 9 Distributions of the cantilever deformation

Table 4
Obtained results for each branch of the 2RPU/UPR mechanism

Table 5
The magnitude of the driving and constraint wrenches under the action of force $ Fx

Table 6
The magnitude of the driving and constraint wrenches under the action of force $ Fy

Table 7
The magnitude of the driving and constraint wrenches under the action of force $ Fz Figure 12The deformation changes in X-direction of 2RPU/UPR PM when the stiffness value changes