 Original Article
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 Published:
GravityBased Kinetostatic Modeling of Parallel Manipulators Using Screw Theory
Chinese Journal of Mechanical Engineering volume 36, Article number: 152 (2023)
Abstract
The pose accuracy of parallel manipulators (PMs) is a key index to measure their performance. Establishing the gravitybased kinetostatic model of a parallel robot provides an important basis for its error composition and accuracy improvement. In this paper, a kinetostatic modeling approach that takes real gravity distribution into consideration is proposed to analyze the influence of gravity on the infinitesimal twist and actuator forces of PMs. First, the duality of the twist screw and constraint wrenches are used to derive the gravityattached constraint wrenches independent of the external load and the limb stiffness matrix corresponding to the kinematicsbased constraint wrenches. Second, the gravity model of the mechanism is established based on the screw theory and the principle of virtual work. Finally, the analytical formulas of the infinitesimal twist and the actuator force of PMs are obtained, and the influences of the external load, platform gravity, and rod gravity on the stiffness of the mechanism are decoupled. The nonoverconstrained 3RPS and overconstrained 2PRUUPR PMs are taken as examples to verify the proposed method. This research proposes a methodology to analyze the infinitesimal deformation of the mechanism under the influence of gravity.
1 Introduction
Compared with the serial mechanism, the parallel manipulator (PM) has better stiffness performance, which is of great significance for the heavyload scenario and the accuracy improvement of the robot [1, 2]. The pose accuracy of PMs is a key index to measure their performance. Establishing the gravitybased kinetostatic model of a parallel robot provides an important basis for its error composition and accuracy improvement. The kinetostatic modeling approaches [3] mainly include the finite element analysis (FEA) method, experimental method, and analytical modeling method, among which the FEA method needs to remeshing for different configurations of the mechanism, and the calculation is timeconsuming [4]. The cost of the experimental method is high and it is difficult to decouple the influence of joint clearance and component elasticity on the stiffness performance of PMs [5].
The analytical kinetostatic modeling method has become a research hotspot of PMs because of its low computational cost. It mainly includes the matrix structure displacement (MSA) method, the virtual joint method (VJM), the screw theory method, and the strain energy method. Deblaise et al. [6] established the stiffness model of the delta PM based on the MSA method, in which the deformation compatibility equation was obtained by using the principle of the total potential energy extreme value. Klimchik et al. [7] established the stiffness model of NaVaRo planar PM using the MSA method with consideration of joint flexibility. Pashkevich et al. [8] described the link flexibility by lumped 6DOF virtual springs and adopted VJM to establish the stiffness model of two translational DOFs of 3PUU and 3PRPaR PMs. Furthermore, Zhao et al. [9] proposed a stiffness modeling method by combining the VJM and MSA, and established stiffness modeling of the 3RRlS reconfigurable PM and 3(3RRlS) reconfigurable seriesPMs. Hu et al. [10] proposed a stiffness modeling method based on the screw theory and basic deformation superposition principle and studied the stiffness performance of the 2RPU+UPR overconstrained PM. Similar to the method in Ref. [10], Zhao et al. [11, 12] established the limb stiffness matrix by mapping the basic deformation to constraint wrenches and then established the stiffness modeling based on the virtual work principle and space force system equilibrium. Yan et al. [13, 14] proposed a strain energy method to establish the stiffness modeling of nonoverconstrained PMs. Yang et al. [5, 15] further expanded Yan's work and proposed an elastostatic stiffness modeling approach for the overconstrained PMs based on the screw theory and strain energy.
In order to improve the accuracy of the kinetostatic model, researchers began to take the mechanism of gravity into account. Lian et al. [16, 17] established the stiffness modeling of a 5DOF PM with the consideration of component gravity as external loads acting on the end reference point. CervantesSánchez et al. [18] presented the static analysis of spatial PMs by means of the virtual work principle with consideration of the gravity of rods and moving platform as the concentrated forces acting on their center of gravity, respectively. Wang et al. [19] presented the compliance analysis of the 3SPR PM with consideration of component gravity and joint/link compliances based on the compliance superposition. Cao et al. [20] derived the stiffness modeling of the overconstrained PMs considering gravity based on the strain energy and virtual work principle. Mei et al. [21] established the gravity compensation modeling of a fiveaxis PM based on the screw theory and compliance superstition principle. Zhao et al. [22] derived the deformation of a 3DOF parallel spindle head in the gravitational field based on the VJM and screw theory, and obtained the constraint wrench caused by link gravity. However, the influence mechanism of gravity on the infinitesimal twist and actuator force was not revealed and the influence of each component’s gravity on the infinitesimal twist was not decoupled in the abovementioned methods.
The main contributions of this work are as follows: (1) the limb gravityattached constrained wrenches independent of the external loads were proposed, and the influence of rod gravity on the actuator forces and elastic deformation corresponding to kinematicsbased constrained wrenches was established; (2) a systematic kinetostatic modeling with consideration of gravity based on the screw theory, strain energy, space force system equilibrium, and virtual work principle was proposed, and the influence of component gravity on the infinitesimal twist of PMs was decoupled.
The rest of this work is structured as follows. Section 2 presents the procedure of the elastostatic stiffness modeling of PMs with consideration of gravity. The case study of a nonoverconstrained PM is presented in Section 3. Section 4 introduces another case study of an overconstrained PM. Finally, the conclusions of this work are drawn in Section 5.
2 Kinetostatic Modeling of PMs with Consideration of Gravity
Figure 1 shows the schematic diagram of PMs with consideration of gravity. The moving platform is connected to the base through n chains, the fixed coordinate frame OXYZ and the moving coordinate frame oxyz are attached to the base and the moving platform, respectively. The assumptions of the modeling are considered as follows to facilitate the interpretation of gravity influence model proposed in this work: (1) Ignore the joint clearance and friction; (2) The moving platform, base, and joints are considered perfectly rigid (the static stiffness model considering joint elasticity can refer to our previous research results [5]); (3) The axial tension, shear, bending, and torsional deformation of the rods and components gravity are considered.
In this paper, the screw theory is used as the mathematical tool to establish the gravity influence model of PMs in the analytical formula. The detailed process is presented as follows.

(1)
Complete limb constraint wrenches with consideration of gravity.
When the component gravity is ignored, the kinematicsbased constraint wrenches of the ith limb J_{ic} = [$_{ic1}, …, $_{ici}, …] based on the kinematic analysis can be obtained by making the reciprocal product with the twist system zero, $_{ici} is the ith constraint force/couple of the ith limb with its intensity W_{ici} (Figure 1). The limb compliance/stiffness matrix corresponding to the J_{ic} can be obtained based on the strain energy and Cartesian theorem, detailed derivation can refer to Refs. [5, 15].
where W_{ic} = [W_{ic1}, …, W_{ici}, …]^{T}. C_{ic} and K_{ic} are the compliance and stiffness matrices corresponding to the J_{ic}, respectively. Δ_{ic} is the elastic deformation corresponding to the J_{ic}.
In general, the rod gravity will do work on the twist screw and generate additional elastic deformation in the direction of the kinematicsbased constraint wrenches. According to the screw theory, the work done by the kinematicsbased constraint wrenches on the limb twist screw is zero.
where ^{i}S_{ij} is the jth twist screw of the ith limb. The upper left symbol i indicates the vector expressed in the limb coordinate frame.
The gravityattached constraint wrenches J_{ig} = [$_{ig1}, …, $_{igk}, …] is generated to balance the gravity; $_{igk} is the kth gravityattached constraint wrench with its intensity is W_{igk}. According to the static equilibrium conditions of the limb, one can have
where ^{i}$_{iq} is the rod gravity wrench with its intensity W_{iq}.
Combining Eqs. (2) and (3), the intensity of the gravityattached constraint wrenches can be obtained by the property that the reciprocal product of the twist system and the constraint wrenches is zero. It is important to note that the gravityattached constraint wrenches are independent of the external load imposed on the moving platform, and only related to the gravity of the rod. In general, the number of gravityattached constraint wrenches is equal to the constraint degree of freedom of the joint at the connection point with the platform minus the number of kinematicsbased constraint wrenches.
Accordingly, the complete limb constraint wrenches combined with kinematicsbased and gravitybased constrained wrenches are given as follows:

(2)
Kinetostatic modeling with consideration of gravity
Figure 2 shows the force diagram of the moving platform with consideration of gravity. In order to simplify the figure, only one kinematicsbased constraint wrench and one gravityattached constraint wrench are provided at each joint. The equilibrium equation of the moving platform with consideration of gravity is given by
where W = W_{e} + W_{gm}, W_{e} = [f^{T}, m^{T}]^{T} is the external load imposed on the moving platform; f and m denote the force and couple respectively; W_{gm} is the gravity load of the moving platform. W_{i} = [W_{ic}, W_{ig}], W_{ig} = [W_{ig1}, …, W_{igj}, …]^{T}.
According to the virtual work principle of the rigid moving platform, one can have
where Δ is the infinitesimal twist of the point o of the moving platform. Δ_{i} is elastic deformation corresponding to W_{i}.
By separating kinematicsbased and gravitybased constraint wrenches, Eq. (6) can be further written as
Similarly, transpose Eq. (5) and multiply both sides by Δ, one can have
By comparing Eqs. (7) and (8), one can have
Accordingly, Eq. (5) can be rewritten as follows:
with
where Δ_{igc} is projection of the elastic deformation caused by rod gravity on the kinematicsbased constraint wrenches.
Rearrange Eq. (10) lead to
where C is the overall compliance matrix of PMs without considering the gravity, namely the inverse of the overall stiffness matrix K. \(\sum\nolimits_{i = 1}^{n} {{\varvec{J}}_{{i{\text{g}}}} {\varvec{W}}_{{i{\text{g}}}} }\) represents the influence of gravityattached constraint wrenches on the infinitesimal twist of the moving platform. \(\sum\nolimits_{i = 1}^{n} {{\varvec{J}}_{{i{\text{c}}}} {\varvec{K}}_{{i{\text{c}}}} {{\varvec{\varDelta}}}_{{i{\text{gc}}}} }\) denotes the influence of the deformation along the J_{ic} caused by rod gravity on the infinitesimal twist of the moving platform. Eq. (12) not only decouples the influence of external load and component gravity on the stiffness performance of PMs, but also the influence mechanism of gravity on the stiffness performance. When the gravity influence is ignored, Eq. (12) degenerates to Δ = CW, which is consistent with the stiffness modeling of the PMs proposed in Ref. [5].
Next, two case studies that include a nonoverconstrained PM and an overconstrained PM are presented to implement the proposed method in this work, wherein, two different approches are presented, one is that all the independent kinematicsbased constraint wrenches act on the connection point between the limb and the moving platform, and the other is that partial independent kinematicsbased constraint spirals act here.
3 Case Study 1: Nonoverconstrained 3RPS PM
Figure 3 shows the 3RPS PM with three DOFs, the moving platform is connected by a spherical joint at A_{i} to the base by a revolute joint at B_{i}. The global coordinate frame and the moving coordinate frame are attached at centroid O of equilateral triangle B_{1}B_{2}B_{3} and centroid o of equilateral triangle A_{1}A_{2}A_{3}, respectively. The X and xaxes along OB_{1} and oA_{1}, respectively, the Z and zaxes are perpendicular to the base and the moving platform upward, respectively. The limb coordinate frame is attached at point B_{i} with its z_{i} and y_{i}axes point in the direction of B_{i}A_{i} and revolute axis, respectively. Structure and material parameters are designed as: radii of the base and moving platform are r_{1} = 300 mm and r_{2} = 200 mm, respectively, diameter of three rods is d = 100 mm, elasticity modulus E = 200 GPa, poisson ratio μ = 0.3, and material density ρ = 7820 kg/m^{3}. Kinematic analysis of the mechanism can be found in Ref. [23]. The geometric constraints are defined as follows. L_{min} ≤ L_{i} ≤ L_{max}, L_{i} is the length of the rod B_{i}A_{i}, L_{min} = 200 mm and L_{max} = 1000 mm denote the minimum and maximum of the ith rod, respectively. α_{i} ≤ α_{max} with α_{i} and α_{max} = 60° denote the angle and the maximum angle of the joints, respectively.
Based on the screw theory, the RPS limb exerts two forces on the moving platform (as shown in Figure 3(b)), one force passes through the point A_{i} and along the direction of the B_{i}A_{i}, and the other force passes through the point A_{i} and parallels to the axis of the revolute axis. The limb compliance matrix corresponding to constraint wrenches can be obtained through strain energy and Castigliano’s theorem.
where G is the shear modulus; A is the cross sectional area, and I is moment of inertia of crosssection.
As shown in Figure 3(b), the work done by the rod gravity on the revolute axis is not equal to zero except for the gravity vector along the rod axis. Based on the screw theory, it is known that the work of the kinematicsbased constraint wrenches on the twist screw is zero. Therefore, a gravityattached constraint wrench is generated to maintain the equilibrium of the rod. Since the spherical joint does not produce constraint couples, the generated gravityattached constraint wrench is a force passing through point A_{i} and parallel to the x_{i}axis.
According to Eq. (3), the equilibrium equation expressed in the limb coordinate frame is given as follows:
where W_{iq} = qL_{i}, ^{i}$_{iq} = [e_{q}, 0.5^{i}B_{i}A_{i}×e_{q}]^{T}, and e_{q} is the unit vector of gravity distribution; ^{i}S_{i1} = [e_{2}, 0, 0, 0]^{T} is the twist screw of the revolute axis with e_{2} = [0, 1, 0]^{T}, and ^{i}$_{ig1} = [e_{1}, ^{i}B_{i}A_{i}×e_{1}]^{T} with e_{1} = [1, 0, 0]^{T}.
According to Eq. (14), W_{ig1} can be obtained as follows:
where q_{ix} is the component of vector q on the x_{i}axis.
According to the geometric constraints of the mechanism, the gravity load q has components only in the x_{i} and z_{i}axes. Accordingly, the elastic deformation on the direction of constraint wrenches caused by the gravity load is given by
where q_{iz} is the component of gravity load q on the x_{i}axis.
Accordingly, the infinitesimal twist of the point o of the moving platform can be obtained by Eq. (12), herein, J_{ig} = [R_{i}e_{1}, oA_{i}×R_{i}e_{1}]^{T}, J_{ic1} = [R_{i}e_{3}, oA_{i}×R_{i}e_{3}]^{T}, J_{ig} = [R_{i}e_{2}, oA_{i}×R_{i}e_{2}]^{T}, R_{i} is the rotation matrix from limb to global coordinate frame, and e_{3} = [0, 0, 1]^{T}.
Furthermore, the actuation force of the ith limb to equilibrium gravity loads can be obtained as follows:
Considering whether the mechanism is rationally symmetric, two configurations are selected to verify the correctness of the proposed method: configuration 1, a rotationally symmetric configuration, L_{1} = L_{2} = L_{3} = 550 mm; configuration 2, an asymmetric configuration, L_{1} = 544.30 mm, L_{2} = 488.24 mm, and L_{3} = 498.10 mm. Table 1 shows the comparison of the infinitesimal twist of point o of the 3RPS PM in the analytical and FEA methods when only gravity is considered. The maximum relative error is less than 0.5%. Table 2 shows the comparison of the intensity of constraint wrenches and actuator forces of the 3RPS PM, the maximum relative error is within 0.7%. The results show the accuracy of the kinetostatic modeling with consideration of gravity proposed in this paper. It is worth noting that due to the symmetry of the mechanism in configuration 1, only the results for limb 1 are given in Table 2. Figures 4 and 5 respectively show the FEA results of Configuration 1 and 2 of the 3RPS PM. It is noteworthy that the moving platform shown in Figure 4 is considered to be elastic with the elasticity modulus close to the rigid body to guarantee the graphics quality.
Figure 6 shows the infinitesimal twist of the point o of the 3RPS PM in the regular workspace with the gravity considered. The maximum linear twist is about 16 μm, and the maximum angular twist is about 0.002°. Figure 7 shows the actuator force of the mechanism with the consideration of gravity, the maximum actuator force of 600 N is required to equilibrium the gravity of the mechanism.
4 Case Study 2: Overconstrained 2PRUUPR PM
Figure 8 shows the 3DOF 3PRUUPR PM, namely a translation along the line perpendicular to the two axes of the Ujoint, a rotation β about the yaxis, and a rotation γ about the Xaxis. The moving platform is connected to the base by two PRU limbs and one UPR limb, global coordinate frame OXYZ, moving coordinate frame oxyz, and limb coordinate frame A_{i}x_{i}y_{i}z_{i} are respectively attached to the base, moving platform, and ith limb. The z_{i}axis along the direction of B_{i}A_{i}, x_{i}− (i = 1,2) and y_{3}axes along the direction of the revolute axis in the ith limb. oA_{1} = oA_{2} = oA_{3} = r_{m} = 250 mm, OB_{3} = r_{b} = 500 mm, A_{1}B_{1} = A_{2}B_{2}= L = 700 mm, OB_{1} = a_{1}, OB_{2} = a_{2}, and A_{3}B_{3} = a_{3}, the diameter of the links are d = 60 mm, material constants are the same as those of the 3RPS PM. More details about the inverse kinematics can refer to Ref. [24].
Figure 9(a) shows the complete constraint wrenches of the PRU limb. When gravity is ignored, the PRU limb exerts three constraint wrenches on the moving platform that includes a force W_{ic1} passing through the point A_{i} and in the direction of B_{i}A_{i}, a force W_{ic2} passing the point A_{i} and in the direction of the revolute axis, a couple W_{ic3} perpendicular to two axes of the universal joint. The compliance/stiffness matrix corresponding to the constraint wrenches can be found in Refs. [15, 25]. When gravity is considered, a gravityattached force that passes through the point A_{i} and in the direction y_{i}axis is necessary to equilibrium the work done by the gravity on the revolute axis.
According to Eq. (14), the intensity of the gravityattached constrained wrench can be obtained as follows:
where q_{iy} is the component of vector q on the y_{i}axis.
According to the geometric constraints of the mechanism, the gravity load q has components only in the y_{i} and z_{i} axes. Accordingly, the elastic deformation on the direction of constraint wrenches caused by the gravity load is given by
The UPR limb exerts a force W_{3c1} along the direction of B_{3}A_{3}, a force W_{3c2} passes through point B_{3} and parallel to the revolute axis, and a couple W_{3c3} on the moving platform when its gravity is ignored. There are two approaches to deal with this issues that the constraint wrench is not directly exerted on the connection point with the moving platform: one is to map the elastic deformation caused by rod gravity to the constraint wrench $_{3c2}; the other is to translate the constraint wrench W_{3c2} acting on the point B_{3} to the point A_{3} and attach a couple W_{3c4} along x_{3}axis, and satisfy W_{3c4} = q_{3}W_{3c2}.
For the scenario 1 of the UPR limb: the compliance/stiffness matrix of the UPR limb corresponding to the kinematicsbased constraint wrenches can be found in Ref. [5]. According to screw theory, the works done by the kinematicsbased constraint wrenches on the twist screw S_{31} and S_{32} of the two axes of the universal joint are zero. Accordingly, the gravityattached constrained wrenches can be obtained based on Eq. (14):
Thus, the projection of the elastic deformation caused by the rod gravity on the kinematicsbased constraint wrenches can be obtained as follows:
where \(d_{{3{\text{g}}y}} = \frac{{W_{{3{\text{g}}2}} a_{3}^{3} }}{3EI} + \frac{{q_{3y} a_{3}^{4} }}{8EI}\) and \(\theta_{{3{\text{g}}x}} =  \frac{{W_{{3{\text{g}}2}} a_{3}^{2} }}{2EI}  \frac{{q_{3y} a_{3}^{3} }}{6EI}\) are the linear displacement deformation of the point A_{3} along the y_{3}axis and the angular displacement deformation along the x_{3}axis caused by the rod gravity, respectively. τ is the unit vector of the constraint couple W_{3c3}.
For the scenario 2 of the UPR limb: Due to the coupling relation between W_{3c4} and W_{3c2}, as well as the linear displacement along the y_{3}axis and the angular displacement along the x_{3}axis, the number of the independent kinematicsbased constraint wrenches is three. Thus, the overall stiffness matrix of the mechanism without considering gravity can be expressed as follows:
with
where D_{3} is the mapping matrix from [W_{3c1}, W_{3c2}, W_{3c3}]^{T} to [W_{3c1}, W_{3c2}, W_{3c3}, W_{3c4}]^{T}. The results of Eq. (22) is essentially consistent with that of Ref. [5]. Actually, the \({\varvec{J^{\prime}}}_{{3{\text{c}}}} {\varvec{D}}_{3}\) in Eq. (22) of approach 2 is consistent with J_{3c} in scheme 1.
Since the coupling relation of W_{3c4} and W_{3c2}, the gravityattached constrained wrenches are consistent with that of scheme 1. Now, the elastic deformation corresponding to the kinematicsbased constraint wrenches caused by rod gravity can be established as follows:
Accordingly, the infinitesimal twist of the point o of the moving platform can be obtained by Eq. (12), herein, J_{ic1} = [R_{i}e_{3}, oA_{i} × R_{i}e_{3}]^{T}, J_{ic2} = [R_{i}e_{1}, oA_{i} × R_{i}e_{1}]^{T}, J_{ic3} = [0, 0, 0, τ]^{T}, J_{ig} = [R_{i}e_{2}, oA_{i} × R_{i}e_{2}]^{T} (i = 1, 2), J_{3c1} = [R_{3}e_{3}, oA_{3} × R_{3}e_{3}]^{T}, J_{3c2} = [R_{3}e_{2}, oB_{3} × R_{3}e_{2}]^{T}, J_{3c3} = [0, 0, 0, τ]^{T}, J_{3g1} = [R_{3}e_{1}, oA_{3} × R_{3}e_{1}]^{T}, J_{3g2} = [R_{3}e_{2}, oA_{3} × R_{3}e_{2}]^{T}, \({\varvec{W}}_{{{\text{gm}}}} = \left[ {{\varvec{G}}_{{\text{m}}} ,\;\frac{1}{3}\sum\limits_{i = 1}^{3} {{\varvec{oA}}_{i} } \times {\varvec{G}}_{{\text{m}}} } \right]\), and G_{m} = [0, 0, ρA_{m}hg], A_{m} and h = 50 mm are the basal area and height of the moving platform. For the approach 2: J'_{3c2} = [R_{3}e_{2}, oA_{3}×R_{3}e_{2}]^{T}, J'_{3c4} = [0, 0, 0, R_{3}e_{1}]^{T}.
Similarly, the actuation force of the ith limb to balance gravity loads can be obtained through Eq. (17).
Two configurations are considered to verify the correctness of the proposed method: Configuration 1, a symmetric configuration, z = 600 mm, β = 0, and γ = 0; configuration 2, an asymmetric configuration, z = 600 mm, β = 5º, and γ = − 6°. Table 3 shows the relative error of infinitesimal twist of point o of the 2PRUUPR PM between the analytical and FEA methods with the consideration of gravity, the maximum relative angular twist error is 5.72% of that around Zaxis, the maximum relative linear twist error is 3.08% of that along Yaxis. Table 4 shows the comparison of the intensity of constraint wrenches and actuator forces of the 2PRUUPR PM, the maximum relative error is within 3.3%. The results show the effectiveness of the kinetostatic modeling with consideration of gravity proposed in this paper. Figures 10 and 11 show the FEA results of Configurations 1 and 2 of the 2PRUUPR PM, respectively.
Figure 12 shows the infinitesimal twist of the point o of the 2PRUUPR PM under the gravity load in the cuboid regular workspace with − 10° ≤ β, γ ≤ 10° and 300 mm ≤ z ≤ 600 mm [5]. The maximum linear twist reaches 26 μm, the maximum angular twist reaches 0.0075º. Figure 13 shows the distribution of the actuator force of the mechanism in the regular workspace under gravity load, the additional maximum actuator force 496 N is required to equilibrium the gravity of the mechanism. The comparison analysis of two cases that include a nonoverconstrained PM and an overconstrained PM shows the rationality of the proposed modeling in this work.
5 Conclusions

(1)
This work proposed a kinetostatic modeling approach for PMs based on the screw theory with the consideration of gravity. Based on the dual property of the twist screw and constraint wrenches, the concept of gravityattached constraint wrenches independent of external loads, as well as gravityattached elastic deformation in the direction of the kinematicsbased constraint wrenches were proposed. The influence of component gravity and external load on the infinitesimal twist of the end of PMs was decoupled. The proposed method is applicable to nonredundant actuated nonoverconstrained and overconstrained PMs.

(2)
The 3RPS PM (a nonoverconstrained PM) and 2PRUUPR PM (an overconstrained PM) were considered as two cases to implement the proposed approach. The maximum relative errors of the linear infinitesimal twist of the moving platform and the actuator force between theoretical and FEA methods for the 3RPS PM are within 0.5% and 0.2%, respectively, and that for the 2PRUUPR PM are less than 3.08% and 1.66%, respectively. An additional actuator force of 600 N is required to balance the gravity of the 3RPS PM, and 496 N is needed in the 2PRUUPR PM. The numerical results demonstrate the accuracy of the proposed gravity modeling, which can be considered as a gravity compensation modeling for the feedforward control of PMs. In future works, experimental research on the error compensation of gravity will be carried out to improve the pose accuracy of parallel robots.
Availability of Data and Materials
The corresponding author can provide MATLAB and ANSYS files to support the work.
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Acknowledgements
The authors would like to thank to Pro. Qinchuan Li of Zhejiang SciTech University, China, for his critical discussion and reading during manuscript preparation.
Funding
Supported by National Natural Science Foundation of China (Grant No. 52275036) and Key Research and Development Project of Jiaxing Science and Technology Bureau of China (Grant No. 2022BZ10004).
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CY contributed the conceptualization, methodology, numerical simulation, and original draft of the paper. FH contributed the visualization and supervision of original data, figures, and code of the paper. WY contributed the simulation work, review and editing of the paper. QC contributed the conceptualization and revised the paper. All authors read and approved the final manuscript.
Authors’ Information
Chao Yang, born in 1982, is currently a Lecturer at College of Mechanical and Electrical Engineering, Jiaxing University, China. He received the B.S. degree from Zhengzhou University of Light Industry, China, in 2005, the M.S. degree in engineering mechanics from Dalian University of Technology, China, in 2009, and a Ph.D. degree in mechanical engineering from Zhejiang SciTech University, China, in 2019. His research interests include kinematics, stiffness, dynamics, and multiobjective optimization of parallel manipulators.
Fengli Huang, born in 1976, is currently a professor at College of Mechanical and Electrical Engineering, Jiaxing University, China. He received his B.S. degree in thermal engineering from Kunming University of Science and Technology, China, in 2000, the M.S. degree in mechanical engineering from Zhejiang University of Technology, China, in 2005, and a Ph.D. degree in mechanical engineering from Tongji University, China, in 2010.
Wei Ye, born in 1988, is currently an associate professor at Faculty of Mechanical Engineering & Automation, Zhejiang SciTech University, China. He received his B.S. and Ph.D. degrees in mechanical engineering from Beijing Jiaotong University, China, in 2010 and 2016, respectively. His research interests include mechanism theory of parallel manipulators and application.
Qiaohong Chen, born in 1976, is currently a professor at Zhejiang SciTech University, China. She received her Ph.D. degree in mechanical engineering from Zhejiang SciTech University, China, in 2012. Her research interests include kinematics and computeraided design of parallel manipulator, and pattern recognition.
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Yang, C., Huang, F., Ye, W. et al. GravityBased Kinetostatic Modeling of Parallel Manipulators Using Screw Theory. Chin. J. Mech. Eng. 36, 152 (2023). https://doi.org/10.1186/s10033023009756
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DOI: https://doi.org/10.1186/s10033023009756