Gravity-Based Kinetostatic Modeling of Parallel Manipulators Using Screw Theory

Abstract

The pose accuracy of parallel manipulators (PMs) is a key index to measure their performance. Establishing the gravity-based 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 gravity-attached constraint wrenches independent of the external load and the limb stiffness matrix corresponding to the kinematics-based 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 non-overconstrained 3RPS and overconstrained 2PRU-UPR 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 heavy-load 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 gravity-based 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 re-meshing for different configurations of the mechanism, and the calculation is time-consuming [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 6-DOF virtual springs and adopted VJM to establish the stiffness model of two translational DOFs of 3-PUU and 3-PRPaR 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 series-PMs. 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 2-RPU+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 non-overconstrained 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 5-DOF PM with the consideration of component gravity as external loads acting on the end reference point. Cervantes-Sánchez 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 3-SPR 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 over-constrained PMs considering gravity based on the strain energy and virtual work principle. Mei et al. [21] established the gravity compensation modeling of a five-axis PM based on the screw theory and compliance superstition principle. Zhao et al. [22] derived the deformation of a 3-DOF 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 above-mentioned methods.

The main contributions of this work are as follows: (1) the limb gravity-attached constrained wrenches independent of the external loads were proposed, and the influence of rod gravity on the actuator forces and elastic deformation corresponding to kinematics-based 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 non-overconstrained 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 O-XYZ and the moving coordinate frame o-xyz 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.

Figure 1

Schematic diagram of PM with consideration of gravity: (a) Force analysis of PM, (b) Complete constraint wrenches of the limb i

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 kinematics-based 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].

$$\left\{ \begin{gathered} {\varvec{C}}_{{i{\text{c}}}} {\varvec{W}}_{{i{\text{c}}}} = {{\varvec{\varDelta}}}_{{i{\text{c}}}} , \hfill \\ {\varvec{K}}_{{i{\text{c}}}} = {\varvec{C}}_{{i{\text{c}}}}^{ - 1} , \hfill \\ \end{gathered} \right.$$

(1)

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 kinematics-based constraint wrenches. According to the screw theory, the work done by the kinematics-based constraint wrenches on the limb twist screw is zero.

$$W_{{i{\text{c}}j}} {}^{i}\varvec{\$ }_{{i{\text{c}}j}} \circ {}^{i}{\varvec{S}}_{ij} = 0,$$

(2)

where ⁱS_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 gravity-attached constraint wrenches J_ig = [$_ig1, …, $_igk, …] is generated to balance the gravity; $_igk is the kth gravity-attached constraint wrench with its intensity is W_igk. According to the static equilibrium conditions of the limb, one can have

$$W_{{i{\text{c}}j}} {}^{i}{\varvec{\$ }}_{{i{\text{c}}j}} \circ {}^{i}{\varvec{S}}_{ij} + W_{{i{\text{q}}}} {}^{i}{\varvec{\$ }}_{{i{\text{q}}}} \circ {}^{i}{\varvec{S}}_{ij} + W_{{i{\text{g}}k}} {}^{i}{\varvec{\$ }}_{{i{\text{g}}k}} \circ {}^{i}{\varvec{S}}_{ij} = 0,$$

(3)

where ⁱ$_iq is the rod gravity wrench with its intensity W_iq.

Combining Eqs. (2) and (3), the intensity of the gravity-attached 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 gravity-attached 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 gravity-attached constraint wrenches is equal to the constraint degree of freedom of the joint at the connection point with the platform minus the number of kinematics-based constraint wrenches.

Accordingly, the complete limb constraint wrenches combined with kinematics-based and gravity-based constrained wrenches are given as follows:

$${\varvec{J}}_{i} = \left[ {{\varvec{J}}_{{i{\text{c}}1}} , \ldots ,{\varvec{J}}_{{i{\text{c}}i}} , \ldots {\varvec{J}}_{{i{\text{g}}1}} , \ldots {\varvec{J}}_{{i{\text{g}}j}} , \ldots } \right].$$

(4)

• (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 kinematics-based constraint wrench and one gravity-attached constraint wrench are provided at each joint. The equilibrium equation of the moving platform with consideration of gravity is given by

$${\varvec{W}} = \sum\limits_{i = 1}^{n} {{\varvec{J}}_{i} {\varvec{W}}_{i} } = \sum\limits_{i = 1}^{n} {{\varvec{J}}_{{i{\text{c}}}} {\varvec{W}}_{{i{\text{c}}}} } + \sum\limits_{i = 1}^{n} {{\varvec{J}}_{{i{\text{g}}}} {\varvec{W}}_{{i{\text{g}}}} } ,$$

(5)

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.

Figure 2

Force analysis diagram of the moving platform with consideration of gravity

According to the virtual work principle of the rigid moving platform, one can have

$${\varvec{W}}^{{\text{T}}} \varvec{\varDelta = }\sum\limits_{i = 1}^{n} {{\varvec{W}}_{i}^{{\text{T}}} }{\varvec{\varDelta}}_{i} ,$$

(6)

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 kinematics-based and gravity-based constraint wrenches, Eq. (6) can be further written as

$${\varvec{W}}^{{\text{T}}} {{\varvec{\varDelta}}} = \sum\limits_{i = 1}^{n} {{\varvec{W}}_{{i{\text{c}}}}^{{\text{T}}} {{\varvec{\varDelta}}}_{{i{\text{c}}}} } + \sum\limits_{i = 1}^{n} {{\varvec{W}}_{{i{\text{g}}}}^{{\text{T}}} {{\varvec{\varDelta}}}_{{i{\text{g}}}} } .$$

(7)

Similarly, transpose Eq. (5) and multiply both sides by Δ, one can have

$${\varvec{W}}^{{\text{T}}} {{\varvec{\varDelta}}} = \sum\limits_{i = 1}^{n} {{\varvec{W}}_{{i{\text{c}}}}^{{\text{T}}} {\varvec{J}}_{{i{\text{c}}}}^{{\text{T}}} {{\varvec{\varDelta}}}} + \sum\limits_{i = 1}^{n} {{\varvec{W}}_{{i{\text{g}}}}^{{\text{T}}} {\varvec{J}}_{{i{\text{g}}}}^{{\text{T}}} {{\varvec{\varDelta}}}} .$$

(8)

By comparing Eqs. (7) and (8), one can have

$$\left\{ \begin{gathered} {{\varvec{\varDelta}}}_{{i{\text{c}}}} = {\varvec{J}}_{{i{\text{c}}}}^{{\text{T}}} {{\varvec{\varDelta}}}, \hfill \\ {{\varvec{\varDelta}}}_{{i{\text{g}}}} = {\varvec{J}}_{{i{\text{g}}}}^\text{T} {{\varvec{\varDelta}}}. \hfill \\ \end{gathered} \right.$$

(9)

Accordingly, Eq. (5) can be rewritten as follows:

$${\varvec{W}} - \sum\limits_{i = 1}^{n} {{\varvec{J}}_{{i{\text{g}}}} {\varvec{W}}_{{i{\text{g}}}} } = \sum\limits_{i = 1}^{n} {{\varvec{J}}_{{i{\text{c}}}} {\varvec{K}}_{{i{\text{c}}}} ({{\varvec{\varDelta}}}_{{i{\text{c}}}} - {{\varvec{\varDelta}}}_{{i{\text{gc}}}} )} ,$$

(10)

with

$${\varvec{W}}_{{i{\text{c}}}} = {\varvec{K}}_{{i{\text{c}}}} \left( {{{\varvec{\varDelta}}}_{{i{\text{c}}}} - {{\varvec{\varDelta}}}_{{i{\text{gc}}}} } \right),$$

(11)

where Δ_igc is projection of the elastic deformation caused by rod gravity on the kinematics-based constraint wrenches.

Rearrange Eq. (10) lead to

$$\left\{ \begin{gathered} {{\varvec{\varDelta}}} = {\varvec{C}}\left( {{\varvec{W}} - \sum\limits_{i = 1}^{n} {{\varvec{J}}_{{i{\text{g}}}} {\varvec{W}}_{{i{\text{g}}}} } + \sum\limits_{i = 1}^{n} {{\varvec{J}}_{{i{\text{c}}}} {\varvec{K}}_{{i{\text{c}}}} {{\varvec{\varDelta}}}_{{i{\text{gc}}}} } } \right), \hfill \\ {\varvec{C}} = {\varvec{K}}^{ - 1} ,\quad {\varvec{K}} = \sum\limits_{i = 1}^{n} {{\varvec{J}}_{{i{\text{c}}}} {\varvec{K}}_{{i{\text{c}}}} {{\varvec{\varDelta}}}_{{i{\text{c}}}}^{{\text{T}}} ,} \hfill \\ \end{gathered} \right.$$

(12)

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 gravity-attached 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 non-overconstrained 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 kinematics-based constraint wrenches act on the connection point between the limb and the moving platform, and the other is that partial independent kinematics-based constraint spirals act here.

3 Case Study 1: Non-overconstrained 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 B₁B₂B₃ and centroid o of equilateral triangle A₁A₂A₃, respectively. The X- and x-axes along OB₁ and oA₁, respectively, the Z- and z-axes 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_iA_i and revolute axis, respectively. Structure and material parameters are designed as: radii of the base and moving platform are r₁ = 300 mm and r₂ = 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/m³. 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_iA_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.

Figure 3

3RPS PM: (a) Schematic diagram, (b) Complete constrained wrenches of RPS limb with consideration of gravity

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_iA_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.

$$C_{{i{\text{c}}}} = \left[ {\begin{array}{*{20}c} {\frac{{L_{i} }}{EA}} & 0 \\ 0 & {\frac{{L_{i} }}{GA} + \frac{{L_{i}^{3} }}{3EI}} \\ \end{array} } \right],$$

(13)

where G is the shear modulus; A is the cross sectional area, and I is moment of inertia of cross-section.

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 kinematics-based constraint wrenches on the twist screw is zero. Therefore, a gravity-attached constraint wrench is generated to maintain the equilibrium of the rod. Since the spherical joint does not produce constraint couples, the generated gravity-attached 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:

$$W_{{i{\text{q}}}} {}^{i}{\varvec{\$ }}_{{i{\text{q}}}} \circ {}^{i}{\varvec{S}}_{i1} + W_{{i{\text{g}}1}} {}^{i}{\varvec{\$ }}_{{i{\text{g}}1}} \circ {}^{i}{\varvec{S}}_{i1} = 0,$$

(14)

where W_iq = qL_i, ⁱ$_iq = [e_q, 0.5ⁱB_iA_i×e_q]^T, and e_q is the unit vector of gravity distribution; ⁱS_i1 = [e₂, 0, 0, 0]^T is the twist screw of the revolute axis with e₂ = [0, 1, 0]^T, and ⁱ$_ig1 = [e₁, ⁱB_iA_i×e₁]^T with e₁ = [1, 0, 0]^T.

According to Eq. (14), W_ig1 can be obtained as follows:

$$W_{{i{\text{g}}1}} = - 0.5q_{ix} L_{i} ,$$

(15)

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

$${{\varvec{\varDelta}}}_{{i{\text{gc}}}} = \left[ {\begin{array}{*{20}c} {\frac{{q_{iz} L_{i}^{2} }}{2EA}} & 0 \\ \end{array} } \right]^{{\text{T}}} ,$$

(16)

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_ie₁, oA_i×R_ie₁]^T, J_ic1 = [R_ie₃, oA_i×R_ie₃]^T, J_ig = [R_ie₂, oA_i×R_ie₂]^T, R_i is the rotation matrix from limb to global coordinate frame, and e₃ = [0, 0, 1]^T.

Furthermore, the actuation force of the ith limb to equilibrium gravity loads can be obtained as follows:

$$W_{{i{\text{a}}}} = W_{{i{\text{c}}1}} + q_{iz} L_{i} .$$

(17)

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, L₁ = L₂ = L₃ = 550 mm; configuration 2, an asymmetric configuration, L₁ = 544.30 mm, L₂ = 488.24 mm, and L₃ = 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.

Table 1 Comparison of infinitesimal twist of point o of the 3RPS PM in the analytical and FEA methods

Table 2 Comparison of the intensity of constraint wrenches and actuator forces of the 3RPS PM in the analytical and FEA methods

Figure 4

FEA results for the configuration 1 of the 3RPS PM: (a) Total deformation, (b) Force reaction at point B₁

Figure 5

FEA results for the configuration 2 of the 3RPS PM: (a) Total deformation, (b) Force reaction at point B₁, (c) Force reaction at point B₂, (d) Force reaction at point B₃

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.

Figure 6

Infinitesimal twist of point o of the 3RPS PM under the influence of gravity: (a) Δ_dx, (b) Δ_dy, (c) Δ_dz, (d) Δ_rx, (e) Δ_ry, (f) Δ_rz.

Figure 7

Actuator force of the 3RPS PM under the influence of gravity: (a) f_1a, (b) f_2a, (c) f_3a

4 Case Study 2: Overconstrained 2PRU-UPR PM

Figure 8 shows the 3-DOF 3PRU-UPR PM, namely a translation along the line perpendicular to the two axes of the U-joint, a rotation β about the y-axis, and a rotation γ about the X-axis. The moving platform is connected to the base by two PRU limbs and one UPR limb, global coordinate frame O-XYZ, moving coordinate frame o-xyz, and limb coordinate frame A_i-x_iy_iz_i are respectively attached to the base, moving platform, and ith limb. The z_i-axis along the direction of B_iA_i, x_i− (i = 1,2) and y₃-axes along the direction of the revolute axis in the ith limb. oA₁ = oA₂ = oA₃ = r_m = 250 mm, OB₃ = r_b = 500 mm, A₁B₁ = A₂B₂= L = 700 mm, OB₁ = a₁, OB₂ = a₂, and A₃B₃ = a₃, 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 8

Schematic diagram of the 2PRU-UPR PM

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_iA_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 gravity-attached 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.

Figure 9

Complete constraint wrenches of limbs of the 2PRU-UPR PM: (a) PRU limbs (i = 1, 2), (b) UPR limb: scenario 1, (c) UPR limb: scenario 2

According to Eq. (14), the intensity of the gravity-attached constrained wrench can be obtained as follows:

$$W_{{i{\text{g}}1}} = - 0.5q_{iy} L_{i} ,\quad (i = 1,\;2),$$

(18)

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

$${{\varvec{\varDelta}}}_{{i{\text{gc}}}} = \left[ {\begin{array}{*{20}c} {\frac{{q_{iz} L_{i}^{2} }}{2EA}} & 0 & 0 \\ \end{array} } \right],\quad (i = 1,\;2).$$

(19)

The UPR limb exerts a force W_3c1 along the direction of B₃A₃, a force W_3c2 passes through point B₃ 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 B₃ to the point A₃ and attach a couple W_3c4 along x₃-axis, and satisfy W_3c4 = q₃W_3c2.

For the scenario 1 of the UPR limb: the compliance/stiffness matrix of the UPR limb corresponding to the kinematics-based constraint wrenches can be found in Ref. [5]. According to screw theory, the works done by the kinematics-based constraint wrenches on the twist screw S₃₁ and S₃₂ of the two axes of the universal joint are zero. Accordingly, the gravity-attached constrained wrenches can be obtained based on Eq. (14):

$$\left\{ \begin{gathered} W_{{3{\text{g}}1}} = - 0.5q_{3x} a_{3} , \hfill \\ W_{{3{\text{g}}2}} = - 0.5q_{3y} a_{3} . \hfill \\ \end{gathered} \right.$$

(20)

Thus, the projection of the elastic deformation caused by the rod gravity on the kinematics-based constraint wrenches can be obtained as follows:

$${{\varvec{\varDelta}}}_{{3{\text{gc}}}} = \left[ {\begin{array}{*{20}c} {\frac{{q_{3z} a_{3}^{2} }}{2EA}} & {d_{{3{\text{g}}y}} + \theta_{{3{\text{g}}x}} a_{3} } & {\theta_{3x} ({\varvec{R}}_{3} {\varvec{e}}_{1} \cdot {{\varvec{\uptau}}})} \\ \end{array} } \right],$$

(21)

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 A₃ along the y₃-axis and the angular displacement deformation along the x₃-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 y₃-axis and the angular displacement along the x₃-axis, the number of the independent kinematics-based constraint wrenches is three. Thus, the overall stiffness matrix of the mechanism without considering gravity can be expressed as follows:

$${\varvec{K}} = \sum\limits_{i = 1}^{2} {{\varvec{J}}_{{i{\text{c}}}} {\varvec{K}}_{{i{\text{c}}}} {\varvec{J}}_{{i{\text{c}}}}^{{\text{T}}} } + {\varvec{J^{\prime}}}_{{3{\text{c}}}} {\varvec{D}}_{3} {\varvec{K}}_{{3{\text{c}}}} {\varvec{D}}_{3}^{{\text{T}}} {\varvec{J^{\prime}}}_{{3{\text{c}}}}^{{\text{T}}} ,$$

(22)

with

$${\varvec{D}}_{3} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & {a_{3} } & 0 \\ \end{array} } \right],$$

(23)

where D₃ 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 gravity-attached constrained wrenches are consistent with that of scheme 1. Now, the elastic deformation corresponding to the kinematics-based constraint wrenches caused by rod gravity can be established as follows:

$${\varvec{\varDelta^{\prime}}}_{{3{\text{gc}}}} = \left[ {\begin{array}{*{20}c} {\frac{{q_{3z} a_{3}^{2} }}{2EA}} & {d_{{3{\text{g}}y}} } & {\theta_{3x} ({\varvec{R}}_{3} {\varvec{e}}_{1} \cdot {{\varvec{\uptau}}})} & {\theta_{{3{\text{g}}x}} a_{3} } \\ \end{array} } \right].$$

(24)

Accordingly, the infinitesimal twist of the point o of the moving platform can be obtained by Eq. (12), herein, J_ic1 = [R_ie₃, oA_i × R_ie₃]^T, J_ic2 = [R_ie₁, oA_i × R_ie₁]^T, J_ic3 = [0, 0, 0, τ]^T, J_ig = [R_ie₂, oA_i × R_ie₂]^T (i = 1, 2), J_3c1 = [R₃e₃, oA₃ × R₃e₃]^T, J_3c2 = [R₃e₂, oB₃ × R₃e₂]^T, J_3c3 = [0, 0, 0, τ]^T, J_3g1 = [R₃e₁, oA₃ × R₃e₁]^T, J_3g2 = [R₃e₂, oA₃ × R₃e₂]^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_mhg], A_m and h = 50 mm are the basal area and height of the moving platform. For the approach 2: J'_3c2 = [R₃e₂, oA₃×R₃e₂]^T, J'_3c4 = [0, 0, 0, R₃e₁]^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 2PRU-UPR PM between the analytical and FEA methods with the consideration of gravity, the maximum relative angular twist error is 5.72% of that around Z-axis, the maximum relative linear twist error is 3.08% of that along Y-axis. Table 4 shows the comparison of the intensity of constraint wrenches and actuator forces of the 2PRU-UPR 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 2PRU-UPR PM, respectively.

Table 3 Comparison of infinitesimal twist of point o of the 2PRU-UPR PM in the analytical and FEA methods

Table 4 Comparison of intensity of constraint screw system and actuator forces of the 2PRU-UPR PM in the analytical and FEA methods

Figure 10

FEA results for the configuration 1 of the 2PRU-UPR PM: (a) Total deformation, (b) Force reaction at point B₁, (c) Force reaction at point B₃

Figure 11

FEA results for the configuration 2 of the 2PRU-UPR PM: (a) Total deformation, (b) Force reaction at point B₁, (c) Force reaction at point B₂, (d) Force reaction at point B₃

Figure 12 shows the infinitesimal twist of the point o of the 2PRU-UPR 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 non-overconstrained PM and an overconstrained PM shows the rationality of the proposed modeling in this work.

Figure 12

Infinitesimal twist of point o of the 2PRU-UPR PM under the influence of gravity: (a) Δ_dx, (b) Δ_dy, (c) Δ_dz, (d) Δ_rx, (e) Δ_ry, (f) Δ_rz

Figure 13

Actuator force of the 2PRU-UPR PM under the influence of gravity: (a) f_1a, (b) f_2a, (c) f_3a

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 gravity-attached constraint wrenches independent of external loads, as well as gravity-attached elastic deformation in the direction of the kinematics-based 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 non-redundant actuated non-overconstrained and overconstrained PMs.

• (2) The 3RPS PM (a non-overconstrained PM) and 2PRU-UPR 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 2PRU-UPR 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 2PRU-UPR 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.