 Original Article
 Open access
 Published:
Evaluation on Configuration Stiffness of Overconstrained 2R1T Parallel Mechanisms
Chinese Journal of Mechanical Engineering volumeÂ 37, ArticleÂ number:Â 62 (2024)
Abstract
Currently, two rotations and one translation (2R1T) threedegreeoffreedom (DOF) parallel mechanisms (PMs) are widely applied in fiveDOF 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. Subsequently, 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 rigidflexible coupled force model for the overconstrained 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 ydirections are larger than the other four mechanisms.
1 Introduction
Studies show that numerous parallel mechanisms (PMs) for diverse applications have been proposed so far. Among different types of lowermobility PMs, the tworotationaldegreesoffreedom and onetranslationaldegreeoffreedom (2R1T) PM are the most important ones. It should be indicated that 3RPS PM [1] is a typical 2R1TPM. 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], fiveDOF parallelserial manipulators based on 3UPSUP and 2UPSUP PMs, and Exechon fiveaxis machining center based on the 2UPRSPR PM [16, 17], which are typical applications of the 2R1TPMs. It is worth noting that R, P, S, and U denote revolute, prismatic, spherical, and universal joints, respectively.
In the type synthesis of the 2R1T PMs, different aspects, including overconstraints [18,19,20], symmetry [5, 21], decoupled motions [22,23,24,25], accompanying motions [26, 27], continuous axes [28,29,30,31] and aÂ number of kinematic joints [32] etc., have been taken into account. However, for the designed PMbased equipment, stiffness is an important index that affects the device performance. The configuration stiffness of a mechanism refers to the endeffector 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 2RPUUPR (U = universal joint, P = prismatic joint, R = revolute joint) overconstrained parallel manipulators. Yang and Li [54] propose a unified method for the elastostatic stiffness modeling of overconstrained parallel manipulators. Ding et al. [55] analyzed the accuracy of an overconstrained Stewart platform with actuation stiffness. Cao et al. [56, 57] obtained the stiffness model of overconstrained 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 Â byÂ 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 closedloop 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.Â [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, theÂ above stiffness indexes are obtained based on the overall stiffness matrix of the mechanism, these indexes do not considerÂ 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 2UPRSPR 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 3UPSUP PM [67] that constitutes the Tricept fiveaxis 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 nonmain 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 fiveaxis 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 overconstrained PM's force analysis. Whatâ€™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.
2 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 yaxis, 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 ydirection, 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 PMbased 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 Î“_{Ï‰} and Î“_{Ï†} are the bending and torsional stiffness performance indices, respectively. Meanwhile, Î“_{Ï‰} can be divided into Î“_{Ï‰}^{x} and Î“_{Ï‰}^{y} along in x and ydirections, respectively. Moreover, Ï‰_{i} and Ï†_{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.
3 Research Object
In the present study, the overconstrained 2R1T parallel mechanism 2RPU/UPR (proposed by Li et al. [29]), 2RPU/UPR/RPR, 3UPS/UP, 2UPU/SP, and 2UPR/SPR are considered as the research object. These mechanisms are schematically presented in Figure 2. It is worth noting that there are two PMs containing three branches and two PMs containing four branches, and the structure of the PMs with the same number of branches are different, thus the constraint wrenches provided by the branches for these five mechanisms are all different. Moreover, these mechanisms consist of the parallel part of four 5DOF hybrid robots [32, 68,69,70] and the classic hybrid robot Exechon [66]. And this section is intended to analyze and compare the configuration stiffness of PMs of these five hybrid robots.
The driving and constraint wrenches exerted to the moving platform by each branch are initially analysed. For the 2RPU/UPR PM, the exerted driving and constraint wrenches to the moving platform are \(\varvec{\$ }_{11}^{r}\), \(\varvec{\$ }_{12}^{r}\), \(\varvec{\$ }_{1}^{a}\), \(\varvec{\$ }_{21}^{r}\), \(\varvec{\$ }_{22}^{r}\), \(\varvec{\$ }_{2}^{a}\), \(\varvec{\$ }_{31}^{r}\), \(\varvec{\$ }_{32}^{r}\), and \(\varvec{\$ }_{3}^{a}\). Moreover, the driving and constraint wrenches for the 2RPU/UPR/RPR PM are \(\varvec{\$ }_{11}^{r}\), \(\varvec{\$ }_{12}^{r}\), \(\varvec{\$ }_{1}^{a}\), \(\varvec{\$ }_{21}^{r}\), \(\varvec{\$ }_{22}^{r}\), \(\varvec{\$ }_{2}^{a}\), \(\varvec{\$ }_{31}^{r}\), \(\varvec{\$ }_{32}^{r}\), \(\varvec{\$ }_{3}^{a}\), \(\varvec{\$ }_{41}^{r}\), \(\varvec{\$ }_{42}^{r}\), and \(\varvec{\$ }_{43}^{r}\). For the 3UPS/UP PM, the driving and constraint wrenches are \(\varvec{\$ }_{1}^{a}\), \(\varvec{\$ }_{2}^{a}\), \(\varvec{\$ }_{3}^{a}\), \(\varvec{\$ }_{41}^{r}\), \(\varvec{\$ }_{42}^{r}\), and \(\varvec{\$ }_{43}^{r}\). For the 2UPR/SPR PM, the driving and constraint wrenches are \(\varvec{\$ }_{11}^{r}\), \(\varvec{\$ }_{12}^{r}\), \(\varvec{\$ }_{1}^{a}\), \(\varvec{\$ }_{21}^{r}\), \(\varvec{\$ }_{22}^{r}\), \(\varvec{\$ }_{2}^{a}\), \(\varvec{\$ }_{31}^{r}\), and \(\varvec{\$ }_{3}^{a}\). Finally, the driving and constraint wrenches of the moving platform for the 2UPU/SP PM are \(\varvec{\$ }_{11}^{r}\), \(\varvec{\$ }_{12}^{r}\), \(\varvec{\$ }_{1}^{a}\), \(\varvec{\$ }_{21}^{r}\), \(\varvec{\$ }_{22}^{r}\), \(\varvec{\$ }_{2}^{a}\), \(\varvec{\$ }_{31}^{r}\), \(\varvec{\$ }_{32}^{r}\), and \(\varvec{\$ }_{3}^{a}\). As shown in the Figure 2, the black arrows indicate the driving wrench while the red arrows indicate the constraint wrench.
Suppose that each branchâ€™s upper and lower connecting links are cylindrical rods with structural parameters given in Table 1, where Î¼ is Poisson's ratio, 2a, and 2b are the length of the triangular base of the moving and fixed platform, respectively. Moreover, Ï† 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 crosssection diameters of the upper and lower connecting links, respectively. And the diameter of P joint is d_{p}, length is l_{p}.
Let Î± and Î² be angled rotating around the axes of two rotational DOFs r_{1} and r_{2}, with initial values of Î±=Î²=0^{Â°}. 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.
4 Configuration Stiffness Analysis and Comparison
4.1 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 branchâ€™s lower rod are constant, while the compliance matrix of the branchâ€™s upper rod is variable, which is related to the change of the branch length. Thus, a coefficient Î»_{i} is introduced, then the ith (i=1,2,3,4) branchâ€™s upper rod compliance matrix C_{is} of the mechanism can be written as:
where Î»_{i} is the length ratio, Î»_{i} = l_{is}/ l_{is0}. And l_{is0} is the upper rodâ€™s length of the ith branch, l_{is} is the initial length of the upper rod in the ith 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 platformâ€™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 ith branchâ€™s end is:
where, F_{i} is the applied force at the end of the ith branch, X_{i} is the total deformation at the end of the ith branch, C_{i} is the whole compliance matrix of the ith 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 ith branch shown in Figure 3 can be expressed as
where \({}^{j}{\varvec{J}}_{i}\) is the force transformation matrix between F_{i} and F_{ij}, which can be expressed as
\({}_{i}^{j} {\varvec{R}}\) 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_{ij}w_{ij} attached at point a_{ij}, \({}_{i}^{j} {\varvec{P}}\) denotes the position vector from point a_{ij} to a_{i} in the frame a_{ij}u_{ij}v_{ij}w_{ij}, \(S({}_{i}^{j} {\varvec{P}})\) is the antisymmetric matrix composed of vector \({}_{i}^{j} {\varvec{P}}\).
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 ith branchâ€™s end can be formulated as:
Supposed that C_{x} represents the compliance matrix of branchâ€™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 \({\varvec{J}}_{{{x}}} = \left[ {\begin{array}{*{20}c} {{\varvec{R}}(x,\uppi /2)} & {} \\ {} & {{\varvec{R}}(x,\uppi /2)} \\ \end{array} } \right]\), \({\varvec{J}}_{{{y}}} = \left[ {\begin{array}{*{20}c} {{\varvec{R}}(y,\uppi /2)} & {} \\ {} & {{\varvec{R}}(y,\uppi /2)} \\ \end{array} } \right]\), \(^{2} {\varvec{J}}_{1} = \left[ {\begin{array}{*{20}c} {{}_{1}^{2} {\varvec{R}}} & 0 \\ {S({}_{1}^{2} {\varvec{P}}){}_{1}^{2} {\varvec{R}}} & {{}_{1}^{2} {\varvec{R}}} \\ \end{array} } \right]\) , \(^{3} J_{1} = \left[ {\begin{array}{*{20}c} {_{1}^{3} R} & 0 \\ {S(_{1}^{3} P)_{1}^{3} R} & {_{1}^{3} R} \\ \end{array} } \right]\).
Then the deformation of the branchâ€™s end along the direction of the constraint force can be expressed as:
where \({\varvec{G}}_{f}^{1}\) is the unit screw system expressed in the frame a_{1}u_{1}v_{1}w_{1}, \({\varvec{G}}_{f}^{1} = \left[ {\begin{array}{*{20}c} {\varvec{\$ }_{11}^{r} } & {\varvec{\$ }_{12}^{r} } & {\varvec{\$ }_{1}^{a} } \\ \end{array} } \right]\).
At the same time, the force F_{1} can be expressed as:
Substituting Eq. (11) into Eq. (10), \({\varvec{X}}_{f}^{1}\) can be expressed as:
Then the stiffness of the R_{1}P_{1}U_{1} branch can be obtained:
In Refs.Â [71, 72], the overconstraint 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 established, and the general analytical expression of this type of mechanism was obtained:
where \({\varvec{G}}_{f}^{F}\) 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 ith 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 Î± and Î² are confined to [âˆ’Â Ï€/6~Ï€/6] and [âˆ’Â Ï€/9~Ï€/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 0.335Â 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 xdirection, each PMâ€™s linear deformation in the xdirection 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 ydirection, each PMâ€™s linear deformation in the ydirection is calculated based on Eq. (1), as shown in Figure 5.
Under the action of an external load with 10 NÂ·m exerted on the origin of the moving coordinate system around the zdirection, each mechanismâ€™s angular deformation in the zdirection is calculated based on Eq. (2), as shown in Figure 6.
4.2 Comparative Analysis of Stiffness and Selection of the Optimal Mechanism

(1)
Comparative analysis of bending stiffness along the xdirection: By observing the average linear deformation of the five parallel mechanisms in the xdirection, 2RPU/UPR/RPR PM has the smallest average deformation in the xdirection, followed by 2RPU/UPR PM, 3UPS/UP PM, 2UPR/SPR PM, and 2UPU/SP PM has the largest linear deformation.

(2)
Comparative analysis of the bending stiffness along the ydirection: By observing the average linear deformation of the five parallel mechanisms in the ydirection, 2RPU/UPR/RPR PM has the smallest average deformation in the xdirection, followed by 2RPU/UPR PM, 3UPS/UP PM, 2UPR/SPR PM, and 2UPU/SP PM has the largest linear deformation.

(3)
Comparative analysis of torsional stiffness in zdirection: By observing the average angular deformation of the five parallel mechanisms in the zdirection, 2RPU/UPR/RPR PM has the smallest average deformation in the zdirection, followed by 3UPS/UP PM, 2RPU/UPR PM, 2UPR/SPR PM, and 2UPU/SP PM has the largest linear deformation.
By comparing the data in Figures 4, 5, 6, the 2RPR/UPR/RPR mechanism obtains better bending stiffness in the x and ydirections and torsional stiffness in the zdirection than the other four PMs. Therefore, the mechanism can preferably be used as the PM part of the fiveDOF hybrid machining robot.
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.
5 Force and Deformation Analysis of the Branches
5.1 Force Analysis of the Branches
Considering redundant and typical constraints, the force analysis of the overconstrained 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 rigidflexible coupling simulation model of overconstrained 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 rigidflexible simulation model of overconstrained PMs and realize the continuous force simulation analysis of the PMs in the whole workspace based on the method proposed in Ref.Â [47].
Accordingly, the rigidflexible coupling model of the overconstrained 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=Ï€D^{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 jointâ€™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 rigidflexible coupling model of the five PMs is established. Figure 10(a)â€“(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 oxyz, the load \(\varvec{\$ }_{F}\)= (10 N 10 N 10 N; 10 N \(\cdot\) m 10 N \(\cdot\) m 10 N \(\cdot\) 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=0~1Â s is the motion time.
Figure 11(a)â€“(c) shows variations of the magnitude of driving and constraint force/couple obtained from the rigidflexible coupling simulations.
In Fig. 11,Â f_{a,i} represents the magnitude of the driving force of the ith branch, f_{r,i} the magnitude of the constraint force of the ith branch, m_{r,i} the magnitudes of the constraint couple of the ith 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 overconstrained PMs. Equation (18) can be expressed as follows.
where \({\varvec{G}}_{f}^{F}\) is the mapping matrix from f to \(\varvec{\$ }_{F}\), \(\varvec{\$ }_{F}\) 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.
When the mechanism is in the initial state (i.e., t= 0 s and Î±=Î²=0Â°), 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.
5.2 Comparison of the Branchâ€™s Constraint Force/Couple
Using the principle of the "layered slice", the average amplitude of each branchâ€™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 branchâ€™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 \(\varvec{\$ }_{Fx}\)= (10 N 0 0; 0 0 0) exerted on the moving platformâ€™s origin.
In Table 6, 2UPU/SP (2) means that the mechanism is at the position where the x and Xaxis intersect, \(\varvec{\$ }_{12}^{r}\) and \(\varvec{\$ }_{22}^{r}\) are 0; 2UPU/SP (1) means that the mechanism is at the position where the x and Xaxis are parallel, \(\varvec{\$ }_{11}^{r}\) and \(\varvec{\$ }_{21}^{r}\) are 0. The position of the 2UPU/SP (1) occupies a small part of the workspace under the designation, so the PMâ€™s branch deformation is mainly determined by the deformation at the 2UPU/SP (2) position.
Table 6 gives the average magnitude of each branchâ€™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 \(\varvec{\$ }_{Fy}\)= (0 10 N 0; 0 0 0) exerted on the moving platformâ€™s origin.
Table 7 gives the average magnitude of each branchâ€™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 \(\varvec{\$ }_{Fz}\)= (0 0 0; 0 0 10 NÂ·mm) exerted on the moving platformâ€™s origin.
In the first column of Tables 5, 6 and 7, the branches 1, 2, and 3 inÂ the horizontal axis for the 2RPU/UPR mechanism represent R_{1}P_{1}U_{1}, R_{2}P_{2}U_{2}, and U_{3}P_{3}R_{3} branches, respectively. Moreover, for the 2RPU/UPR/RPR mechanism, the branches 1, 2, 3, and 4 represent R_{1}P_{1}U_{1}, R_{2}P_{2}U_{2}, U_{3}P_{3}R_{3}, and R_{4}P_{4}R_{4} branches, respectively. For the 3UPS/UP mechanism, the branches 1, 2, 3, and 4 represent U_{1}P_{1}S_{1}, U_{2}P_{2}S_{2}, U_{3}P_{3}S_{3}, and U_{4}P_{4} branches, respectively. For the 2UPR/SPR mechanism, the branches 1, 2, and 3 represent U_{1}P_{1}R_{1}, U_{2}P_{2}R_{2}, and S_{3}P_{3}R_{3} branches, respectively. For the 2UPU/SP mechanism, the branches 1, 2, and 3 represent U_{11}P_{1}U_{12}, U_{21}P_{2}U_{22}, and S_{3}P_{3} branches, respectively.
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.

1.
Comparative analysis of the driving and constraint forces/couples of branches under the action of external force \(\varvec{\$ }_{Fx}\).
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 \(\varvec{\$ }_{Fx}\) is applied on the moving platform origin, the constraint force mainly resisting the external force for the 2RPU/UPR PM is \(\varvec{\$ }_{31}^{r}\), the constraint forces mainly resisting the external force for the 2RPU/UPR/RPR PM are \(\varvec{\$ }_{31}^{r}\) and \(\varvec{\$ }_{41}^{r}\), the constraint force mainly resisting the external force for the 3UPS/UP PM is \(\varvec{\$ }_{41}^{r}\), the constraint forces mainly resisting the external force for the 2UPR/SPR PM are \(\varvec{\$ }_{11}^{r}\), \(\varvec{\$ }_{21}^{r}\) and \(\varvec{\$ }_{31}^{r}\), the constraint forces mainly resisting the external force for the 2UPU/SP PM are \(\varvec{\$ }_{11}^{r}\), \(\varvec{\$ }_{21}^{r}\) , \(\varvec{\$ }_{31}^{r}\) and \(\varvec{\$ }_{32}^{r}\). 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 Xdirection.

2.
Comparative analysis of the driving and constraint forces/couples of branches under the action of external force \(\varvec{\$ }_{Fy}\).
It can be seen from Table 6 that when the same external force \(\varvec{\$ }_{Fy}\) is applied, the constraint forces that mainly resist this external force for the 2RPU/UPR PM are \(\varvec{\$ }_{11}^{r}\) and \(\varvec{\$ }_{21}^{r}\), the constraint forces that mainly resist this external force for the 2RPU/UPR/RPR PM are \(\varvec{\$ }_{11}^{r}\),\(\varvec{\$ }_{21}^{r}\) and \(\varvec{\$ }_{42}^{r}\), and the constraint force that mainly resist this external force for the 3UPS/UP PM is \(\varvec{\$ }_{42}^{r}\), the constraint forces that mainly resist this external force for the 2UPR/SPR PM are \(\varvec{\$ }_{11}^{r}\) and \(\varvec{\$ }_{21}^{r}\), and the constraint forces that mainly resist this external force for the 2UPU/SP PM are \(\varvec{\$ }_{11}^{r}\), \(\varvec{\$ }_{21}^{r}\) and \(\varvec{\$ }_{31}^{r}\). 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 Ydirection.

3.
Comparative analysis of the driving and constraint forces/couples of branches under the action of external force \(\varvec{\$ }_{Fz}\).
It can be seen from Table 7 that when the same external force \(\varvec{\$ }_{Fz}\) is applied, the constraint forces that mainly resist this external force for the 2RPU/UPR PM are \(\varvec{\$ }_{12}^{r}\), \(\varvec{\$ }_{22}^{r}\), \(\varvec{\$ }_{31}^{r}\) and \(\varvec{\$ }_{32}^{r}\), the constraint forces that mainly resist this external force for the 2RPU/UPR/RPR PM are \(\varvec{\$ }_{12}^{r}\), \(\varvec{\$ }_{22}^{r}\), \(\varvec{\$ }_{31}^{r}\), \(\varvec{\$ }_{32}^{r}\) and \(\varvec{\$ }_{43}^{r}\), and the constraint force that mainly resist this external force for the 3UPS/UP PM is \(\varvec{\$ }_{43}^{r}\), the constraint forces that mainly resist this external force for the 2UPR/SPR PM are \(\varvec{\$ }_{11}^{r}\), \(\varvec{\$ }_{12}^{r}\), \(\varvec{\$ }_{12}^{r}\), \(\varvec{\$ }_{22}^{r}\) and \(\varvec{\$ }_{31}^{r}\), and the constraint forces that mainly resist this external force for the 2UPU/SP PM are \(\varvec{\$ }_{11}^{r}\), \(\varvec{\$ }_{12}^{r}\), \(\varvec{\$ }_{21}^{r}\), \(\varvec{\$ }_{22}^{r}\), \(\varvec{\$ }_{31}^{r}\) and \(\varvec{\$ }_{32}^{r}\).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 Zdirection.
Therefore, the mechanism 2RPU/UPR/RPR can preferably be used as the PM part of the fiveDOF hybrid machining robot, which is consistent with the analysis results in Section 4.
5.3 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 \(\varvec{\$ }_{F}\)= (10 N 10 N 10 N; 10 N Â·m 10 N Â·m 10N Â·m) exerted on the moving platform origin. When h=0.335 m, the average deformation of the parallel mechanism in the Xdirection 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 Ydirection 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 Zdirection in the whole workspace is as shown in Figure 14.
It can be seen from the observation that the Xdirection 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 Xdirection deformation of the parallel mechanism is. The Ydirection 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 Ydirection deformation of the parallel mechanism is. The Zdirection 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 Zdirection 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 materials with higher strength are used to manufacture these parts.
6 Simulation Verification
In order to verify the correctness and rationality of the proposed stiffness performance indices, considering the fiveDOF hybrid processing robot as the application target, simple engineering threedimensional 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 NÂ·m are exerted along the coordinate system's three coordinate axesÂ x,Â y,Â and zÂ on the moving platform. The stiffness simulation of the PMs of the five hybrid robots is performed. Figures 15, 16, 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 zaxis, with respect to the variable Î±, Î², and d, are shown in Figures 20, 21, 22, respectively. Where Î±, Î², and d represent rotation angle around the xaxis, rotation angle around the yaxis, and distance from point o to O, respectively. In Figure 20, d and Î² are kept at a constant, that is, d=290 mm and Î²=0. In Figure 21, d and Î± are kept at a constant, that is, d=290 mm and Î±=0. In Figure 22, Î± and Î² are kept at a constant, that is, Î±=0 and Î²=0.
It can be seen from Figures 20, 21, 22 that the 2RPR/UPR/RPR mechanism has better bending stiffness in the x and ydirections and torsional stiffness in the zdirection than the other four PMs, consistent with the theoretical analysis result. Therefore, the stiffness evaluation indices and methods are rational.
7 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 stiffness of each branch affecting the configuration stiffness are evaluated, and their specific values are defined. Then, an external load is applied, and the bending and torsional deformation of each PM is calculated. The results obtained from these calculations are utilized to evaluate the parallel mechanism's bending and torsional stiffness performance.

(2)
Based on simulations performed in ADAMS and ANSYS environments, a practical solution is proposed to solve the nondynamic problem of the mechanism, including the prismatic joints. This approach introduces the concept of modal files containing multiinterface nodes. By discretizing the rigid areas within the flexible body, the dynamic behavior of the model is simulated, resulting in a rigidflexible hybrid model for the overconstrained PM. This model enables continuous force simulation analysis of the overconstrained PM with P joints throughout the entire workspace.

(3)
Based on the stiffness evaluation indices and methods, the configuration stiffness performance of the five 2R1T PMs is compared and analyzed. The 2RPU/UPR/RPR mechanism has relatively good bending and torsional stiffness preferred as the PM part of the fiveaxis hybrid machining robot. Additionally, the comparison results demonstrate that the stiffness of fourbranch PMs is not necessarily greater than threebranch PMs.
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.
Data availability
The data that support the findings of this study are available from the corresponding author, [Xu Y D], upon reasonable request.
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Supported by National Natural Science Foundation of China( GrantÂ Nos. 51875495, U2037202), and Hebei ProvincialÂ Science and Technology Project (Grant No.Â 206Z1805G).
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Ma, X., Xu, Z., Xu, Y. et al. Evaluation on Configuration Stiffness of Overconstrained 2R1T Parallel Mechanisms. Chin. J. Mech. Eng. 37, 62 (2024). https://doi.org/10.1186/s10033024010451
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DOI: https://doi.org/10.1186/s10033024010451