Hybrid-model-based stiffness analysis of a three-revolute-prismatic-spherical parallel kinematic machine

Abstract

A three-revolute-prismatic-spherical parallel kinematic machine is proposed as an alternative solution for high-speed machining tool due to its high rigidity and high dynamics. Considering the parallel kinematic machine module as a typical compliant parallel mechanism, whose three limb assemblages have bending, extending and torsional deflections, this article proposes a hybrid modeling methodology to establish an analytical stiffness model for the three-revolute-prismatic-spherical device. The developed analytical model is further used to evaluate the stiffness mapping of the three-revolute-prismatic-spherical module over a given work plane which is then validated by experimental tests. The simulations and experiments indicate that the present hybrid methodology can predict the three-revolute-prismatic-spherical parallel kinematic machine’s stiffness in a quick and accurate manner. The solution for eigenvalue problem of the stiffness matrix leads to the stiffness characteristics of the parallel module including eigenstiffnesses and the corresponding eigenscrews as well as the equivalent screw spring constants. Based on the eigenscrew decomposition, the parallel kinematic machine is physically interpreted as a rigid platform suspending by six screw springs. The minimum, maximum and average of the screw spring constants are chosen as indices to assess the three-revolute-prismatic-spherical parallel kinematic machine’s stiffness performance. The distributions of the proposed indices throughout the workspace reveal a strong dependency on the mechanism’s configurations. At the final stage, the effects of some design parameters on system stiffness characteristics are investigated with the purpose of providing useful information for the conceptual design and performance improvement of the parallel kinematic machine.

Introduction

High-speed machining (HSM) has been recognized as one of the key manufacturing technologies for high productivity and throughput. However, the definition of HSM is not easy since the actual cutting speed that can be achieved depends on the work material, the type of cutting operation, the cutting tool used and so on. ¹ In compliance with modern knowledge, the Institute of Production Engineering and Machine Tools (PTW) at the Darmstadt University of Technology defined high-speed machining as being where conventional cutting speeds of a particular material are exceeded by a factor of 5–10. ² From the productivity point of view, HSM is not only a technical index but also an economical index, which can achieve higher productivity and throughput. Compared with their counterparts of serial machine tools, parallel kinematic machines (PKMs) with lower mobility have found their promising potentials in HSM due to the merits of simple structure, high rigidity, better accuracy, good reconfigurability and easy controlling. For example, machining an aircraft wing over 30 m long would require a huge gantry five-axis machine tool with tons of weight and large footprint. However, a much more compact device with comparable machining accuracy can be achieved by using a parallel arrangement. This has been fully exemplified by the commercial success of the Sprint Z3 Head and the Tricept robots applied in aeronautical and automobile industries for HSM tasks.^3–5 Other propositions of applying PKMs for HSM can also be traced in recent publications.^6–9

Inspired by the successful application of Sprint Z3 Head, a similar three-degree-of-freedom (3-DOF) PKM module was proposed with one translational and two rotational capabilities, whose topological structure is a three-revolute-prismatic-spherical (3-RPS) parallel mechanism. ¹⁰ Added by an x–y motion, this proposed design can be used as a multiple-axis spindle head to form a hybrid five-axis HSM unit. When designed for HSM applications, such a kind of PKM module needs to meet the requirements of high rigidity and high positioning accuracy, which makes stiffness become the most overwhelming concern in the design stage. The evaluation of a PKM’s stiffness performance involves two kernel issues, that is, stiffness modeling and evaluation indices.

For the first issue, the substantial work is how to establish a mathematical model to reflect the platform’s ability of resisting deflections subject to external loads. And such an ability of a PKM can be generally described by a 6 × 6 stiffness matrix, which relates the vector of compliant deformations of the PKM to an external static wrench applied at the end-effector. Unlike the hexapod parallel manipulator, whose stiffness matrix can be straightforwardly derived by considering the compliances of each component, the derivation of the overall stiffness matrices for lower mobility parallel manipulators, such as a 3-RPS PKM module, is quite difficult due to their kinematic and complex structural features. ¹¹

Numerous efforts can be traced toward the stiffness modeling of lower mobility parallel manipulators in the past decades. Among all these efforts, the finite element method (FEM),^12,13 the matrix structure method (MSM),^14,15 the virtual joint method (VJM)^16–18 and the screw-based method (SBM)^19–22 are the most commonly used approaches. For example, Huang et al. ¹⁴ proposed a stiffness model for a tripod-based PKM by decomposing the overall system into two separate substructures and formulating the stiffness expressions of each substructure with virtual work principle. A similar model of the 3-DOF CaPaMan parallel manipulator was established by Ceccarelli and Carbone, ¹⁸ which considered the kinematic and static features of three legs in view of the motions of each joint and link. Considering the compliances of actuations and constraints, Li and Xu ²⁰ proposed an intuitive method based on an overall Jacobian to formulate the stiffness matrix of a three-prismatic–universal–universal (3-PUU) translational PKM. And the overall stiffness matrix of the lower mobility parallel manipulator could be derived intuitively. Following the same track, Huang et al. ²¹ proposed a stiffness modeling approach for lower mobility parallel manipulators by using the generalized Jacobian. Dai and Ding ²² proposed a comprehensive analytical compliance model for a three-legged rigidly connected platform device. By solving the eigenvalue problem, the eigencompliances and eigentwists of the flexible parallel mechanism were obtained and the effects of design parameters on compliance property were outlined.

As evidenced by the literatures, the challenge of stiffness modeling is how the proposed mathematical model can predict the stiffness performance over the entire workspace precisely and quickly. This is of great importance in that in most circumstances a PKM’s stiffness is strongly dependent on the manipulator’s configurations, which makes an FE model with large DOFs unacceptable due to high computational cost. On the contrary, an analytical stiffness model may be computationally efficient but lack accuracy when used to predict the PKM’s stiffness. Accordingly, it might be an applaudable choice to develop a hybrid stiffness model that combines the benefits of accuracy of the FEM and concision of the analytical method. Hence, such a kind of hybrid model can be used to predict the stiffness characteristics of a PKM throughout its entire workspace in a quick and accurate manner. This can be useful to evaluate whether the design is satisfied with the stiffness requirements or even further to perform an optimal design with the stiffness considered especially in the design stage. In addition, it is necessary to investigate the stiffness behavior of a PKM at specified configurations to have an in-depth understanding of its stiffness behavior.

As to the second issue in the stiffness design process of a PKM, the fundamental consideration is to provide a proper or suitable index to evaluate its stiffness performance. The authors’ review of previous investigations shows that several performance indices have been proposed for the stiffness evaluation of PKMs. The simplest one is to use stiffness factors, that is, the entries of the stiffness matrix. This, however, is only applicable to evaluate a PKM with diagonal stiffness matrix.^14,23 Additionally, the trace and/or determinant of a stiffness matrix has been adopted to assess the stiffness of parallel manipulators.^18,24 However, a high value of the determinant or trace does not definitely mean good stiffness performance of a PKM. For example, when a manipulator possesses very low rigidity in one direction but with very high stiffness in other directions, the determinant or trace may be very high but the low stiffness direction may prohibit its application as machining tools. Furthermore, the condition number of the stiffness matrix has been introduced, and then a global stiffness index defined as the inverse of the condition number of the stiffness matrix integrated over the reachable workspace and divided by the workspace volume is presented to assess the stiffness of a 3-DOF spherical parallel manipulator. ²⁵ The proposition of condition number of a stiffness matrix can indicate the ill-conditioning of a PKM but cannot provide direct information on stiffness values.

Besides the aforementioned indices, there is another way to evaluate the stiffness performance of a PKM, that is, to use the eigenvalue of the stiffness matrix. It has been verified that the stiffness is bounded by the minimum and maximum eigenvalues of the stiffness matrix. ²⁶ If a PKM is designed for machining tool application, the minimum stiffness throughout the workspace should be over a threshold to ensure the machining accuracy. From this point of view, it is reasonable to adopt the minimum and maximum eigenvalues of stiffness matrix as indices to evaluate the stiffness performance for the proposed 3-RPS PKM design.

Considering the proposed design as a compliant parallel mechanism where the three RPS limbs are equivalent to three sets of springs, which have bending, extending and torsional deflections, this article presents a comprehensive hybrid stiffness model for the 3-RPS PKM module and an in-depth investigation of the system stiffness properties. The analysis is then extended to the effects of design parameters on the system rigidity in order to provide a designer with useful information in the conceptual design stage.

Kinematic description

A computer-aided design (CAD) model for the 3-RPS PKM is shown in Figure 1.

Figure 1.

Structure of the 3-RPS design.

As shown in Figure 1, the 3-RPS PKM module consists of a moving platform, a fixed base and three identical kinematic limbs. Each limb connects the fixed base to the moving platform by a revolute (R) joint which is followed by a prismatic (P) joint and a spherical (S) joint in sequence, where the P joint is driven by a lead-screw linear actuator. An electrical spindle is mounted on the platform to implement high-speed milling. Independently driven by three servomotors, one translation along the z-axis and two rotations about the x-axes and y-axes can be achieved. The detailed descriptions of mechanical features of the 3-RPS design were addressed in our previous work. ²⁷

For the purpose of analysis, a schematic diagram of the 3-RPS PKM is depicted in Figure 2. Herein, A_i and B_i (i = 1–3) are the centers of spherical and revolute joints, respectively; C_i denotes the rear end of the limb; ΔA₁A₂A₃ and ΔB₁B₂B₃ are assumed to be equilateral.

Figure 2.

Schematic diagram of the 3-RPS PKM in Figure 1.

In order to work out a suitable formulation, Cartesian coordinate systems are set as follows. The reference coordinate system B-xyz is attached at the center point B of the fixed base, with the x-axis parallel to B₂B₃ and the z-axis normal to ΔB₁B₂B₃; the body-fixed coordinate system A-uvw is placed at the center point A of the moving platform, with the u-axis parallel to A₂A₃ and w normal to ΔA₁A₂A₃; the limb reference frame B_i-x_iy_iz_i is established at the center point B_i of the ith revolute joint, with x_i and z_i coincident with the axes of revolute joints and limbs, respectively; the centroid reference frame $N_i^c-x_i^c{y}_i^c{z}_i^c$ is set at the mass center of the ith limb body and parallel to B_i-x_iy_iz_i. For clarity, only one centroid reference frame in limb 1 is depicted in Figure 2.

Assume the transformation matrix R₀ of the frame A-uvw with respect to the frame B-xyz can be formulated as

$$R_0=\left[\begin{array}{ccc}\mathrm{c}\psi \mathrm{c}\varphi -\mathrm{s}\psi \mathrm{c}\theta \mathrm{s}\varphi & -\mathrm{c}\psi \mathrm{s}\varphi -\mathrm{s}\psi \mathrm{c}\theta \mathrm{c}\varphi & \mathrm{s}\psi \mathrm{s}\theta \\ \mathrm{s}\psi \mathrm{c}\varphi +\mathrm{c}\psi \mathrm{c}\theta \mathrm{s}\varphi & -\mathrm{s}\psi \mathrm{s}\varphi +\mathrm{c}\psi \mathrm{c}\theta \mathrm{c}\varphi & -\mathrm{c}\psi \mathrm{s}\theta \\ \mathrm{s}\theta \mathrm{s}\varphi & \mathrm{s}\theta \mathrm{c}\varphi & \mathrm{c}\theta \end{array}\right]$$

(1)

where “s” and “c” denote “sin” and “cos” functions, respectively; ψ, θ and ϕ are the Euler angles in terms of precession, nutation and rotation, respectively.

Vectors of points A and B_i measured in the B-xyz frame are defined as ${\mathit{V}}_A^B$ and ${\mathit{V}}_{B_i}^B$ , respectively. And vector of point A_i measured in the A-uvw frame is denoted as ${\mathit{V}}_{A_i}^A$ . They can be expressed as

$$V_{A_i}^A=r_{\mathrm{P}}\left(\begin{array}{c}\cos {\beta }_i\\ \sin {\beta }_i\\ 0\end{array}\right),V_{B_i}^B=r_{\mathrm{b}}\left(\begin{array}{c}\cos {\beta }_i\\ \sin {\beta }_i\\ 0\end{array}\right),V_A^B=\left(\begin{array}{c}{p}_x\\ p_y\\ p_z\end{array}\right)$$

(2)

where r_p and r_b are the radii of the moving platform and the fixed base, respectively; ${\beta }_i=2\pi (i-1)/3-\mathrm{\pi }/2$ are the position angles of the joints; p_x, p_y and p_z are coordinates of point A measured in B-xyz.

Supposing d_i denotes the distance between A_i and B_i and ${\mathit{s}}_{0i}$ is the unit vector of the ith limb, according to Figure 2 the position vector of point A_i measured in the B-xyz can be given as

$${\mathit{V}}_{A_i}^B={\mathbf{R}}_0{\mathit{V}}_{A_i}^A+{\mathit{V}}_A^B={\mathit{V}}_{B_i}^B+d_i{\mathit{s}}_{0i}$$

(3)

Considering the constrains of revolute joint and taking ψ, θ and p_z as independent coordinates, one can obtain the following

$$\{\begin{array}{c}\hfill p_x=0.5r_{\mathrm{p}}(1-\cos \theta )\sin 2\psi \hfill \\ \multicolumn{1}{c}{p_y=0.5r_{\mathrm{p}}(1-\cos \theta )\cos 2\psi }\\ \multicolumn{1}{c}{\varphi =-\psi }\end{array}$$

(4)

Equation (4) represents the parasitic motion of the 3-RPS mechanism. Thus, the inverse position analysis can be conducted by using the following expressions

$$d_i=\Vert {\mathit{V}}_{A_i}^B-{\mathit{V}}_{B_i}^B\Vert$$

(5)

$${\mathit{s}}_{0i}=\frac{{\mathit{V}}_{A_i}^B-{\mathit{V}}_{B_i}^B}{d_i}=\left(\begin{array}{c}\hfill s_{0\mathrm{ix}}\hfill \\ \multicolumn{1}{c}{s_{0\mathrm{iy}}}\\ \multicolumn{1}{c}{s_{0\mathrm{iz}}}\end{array}\right)$$

(6)

Further derivation can give the transformation of the frame B_i-x_iy_iz_i with respect to the frame B-xyz as

$${\mathbf{R}}_i=\mathbf{R}(z,{\vartheta }_i)\mathbf{R}(x,{\gamma }_i)$$

(7)

where R represents the rotation transformation; ${\vartheta }_i$ and ${\gamma }_i$ ( $i=1,2,3$ ) are the rotating angles about z- and x-axes, respectively, and there exist

$${\vartheta }_i=2(i-1)\frac{\pi }{3},{\gamma }_i=\arctan (-\frac{s_{0\mathrm{iy}}}{s_{0\mathrm{iz}}\cos {\vartheta }_i})$$

(8)

Stiffness modeling

This section aims to formulate the stiffness expression of an individual limb as well as that of the PKM system by considering all significant compliances simultaneously. The considered compliances include bending, extension and torsion deflections of the limb body and the deflections of the prismatic, spherical and the revolute joints.

Assumptions

For the convenience of analytical derivation, the following hypotheses and approximations are made:

1. The fixed base and moving platform are treated as rigid bodies due to their relatively high rigidities.

2. The limb body is modeled as a continuous elastic hollowed spatial beam with non-uniform cross sections according to its structural features.

3. The revolute and spherical joints are simplified into virtual lumped springs with equivalent stiffness at their geometric centers.

4. The dampings and frictions in all kinematic pairs are neglected although they can be added to the governing equations with further efforts.

Equilibrium equations of an individual RPS limb assembly

Figure 3 shows the mechanical design of an RPS limb in the 3-RPS PKM module.

Figure 3.

Assembly of an RPS limb.

According to the assembling relationships and structural features of the limb, one can classify all the components in an individual RPS limb into three categories when considering compliance modeling: (1) the limb body, (2) the revolute joint (including the lead-screw assembly) and (3) the spherical joint. Consequently, the compliance of an entire RPS limb can be modeled as a serial combination of the aforementioned three compliant components.

As can be seen from Figure 3, the cross sections of the limb body are non-uniform. To be specific, the cross section of segment A_iD_i of the limb body is in a cylinder form with gradually changing dimensions. The topological form of the cross section is depicted as I. Accordingly, the cross sections of segments D_iE_i, E_iF_i and F_iC_i are labeled as II, III and IV, respectively. By considering the structural features of the limb body, the limb body is modeled as a spatial beam with non-uniform cross sections. As mentioned above, the revolute and spherical joints are represented as virtual lumped springs with equivalent stiffness at their geometric centers. Thus, the RPS assembly can be modeled as a spatial beam supported by two sets of lumped springs as shown in Figure 4. Herein, k_sxi, k_syi and k_szi are linear stiffness coefficients of the ith spherical joint; k_rxi, k_ryi, k_rzi and k_rui, k_rvi, k_rwi are linear stiffness coefficients and angular stiffness coefficients of the ith revolute joint in the ith limb assembly, respectively.

Figure 4.

Simplified force diagram of the ith limb.

The following will introduce the computation of compliances of the revolute and spherical joints, with which the stiffness coefficients of k_sxi, k_syi, k_szi and k_rxi, k_ryi, k_rzi, k_rui, k_rvi, k_rwi can be obtained.

Compliance of revolute joint assembly

As shown in Figure 3, the revolute joint consists of two R joint bearings, a closed frame, two parallel guideway assemblies and a lead-screw assembly containing a front bearing, a rear bearing, a lead screw and a nut. Supposing ${\mathbf{C}}_{\mathrm{r}i}^{\mathrm{b}}$ , ${\mathbf{C}}_{\mathrm{r}i}^{\mathrm{f}}$ , ${\mathbf{C}}_{\mathrm{r}i}^{\mathrm{g}}$ and ${\mathbf{C}}_{\mathrm{r}i}^{\mathrm{l}}$ are the compliance of the R joint bearings, the closed frame, the guideway assembly and the lead-screw assembly, respectively, the compliance of the revolute joint assembly C_ri can be calculated through the superposition of each part as addressed. The expression of C_ri can be formulated as

$${\mathbf{C}}_{\mathrm{r}i}={\mathbf{C}}_{\mathrm{r}i}^{\mathrm{b}}+{\mathbf{C}}_{\mathrm{r}i}^{\mathrm{f}}+{(\frac{1}{{\mathbf{C}}_{\mathrm{r}i}^{\mathrm{g}}}+\frac{1}{{\mathbf{C}}_{\mathrm{ri}}^{\mathrm{l}}})}^{-1}$$

(9)

The R joint bearing provides constraints in five directions and only allows the rotation about its axis. Hence, the compliance matrix of the R joint bearing ${\mathbf{C}}_{\mathrm{r}i}^{\mathrm{b}}$ can be expressed as

$${\mathbf{C}}_{\mathrm{r}i}^{\mathrm{b}}=\mathrm{diag}(\begin{array}{cccccc}\hfill c_{\mathrm{r}\mathrm{xi}}^{\mathrm{b}}\hfill & \hfill c_{\mathrm{r}\mathrm{yi}}^{\mathrm{b}}\hfill & \hfill c_{\mathrm{r}\mathrm{zi}}^{\mathrm{b}}\hfill & \hfill c_{\mathrm{r}\mathrm{ui}}^{\mathrm{b}}\hfill & \hfill c_{\mathrm{r}\mathrm{vi}}^{\mathrm{b}}\hfill & \hfill c_{\mathrm{r}\mathrm{wi}}^{\mathrm{b}}\hfill \end{array})$$

(10)

where $c_{\mathrm{r}\mathrm{ui}}^{\mathrm{b}}=0$ . $c_{\mathrm{r}\mathrm{xi}}^{\mathrm{b}}$ , $c_{\mathrm{r}\mathrm{yi}}^{\mathrm{b}}$ and $c_{\mathrm{r}\mathrm{zi}}^{\mathrm{b}}$ are the three translational compliance (reciprocal of stiffness) coefficients along three Cartesian axes of x, y and z; $c_{\mathrm{r}\mathrm{ui}}^{\mathrm{b}}$ , $c_{\mathrm{r}\mathrm{vi}}^{\mathrm{b}}$ and $c_{\mathrm{r}\mathrm{wi}}^{\mathrm{b}}$ are the three rotational compliance about three Cartesian axes of x, y and z. The coefficients in equation (10) can be determined according to the bearing handbook and installation conditions.

Similarly, the compliance matrix of the closed frame ${\mathbf{C}}_{\mathrm{r}i}^{\mathrm{f}}$ can be expressed as

$${\mathbf{C}}_{\mathrm{r}i}^{\mathrm{f}}=\mathrm{diag}(\begin{array}{cccccc}\hfill c_{\mathrm{r}\mathrm{xi}}^{\mathrm{f}}\hfill & \hfill c_{\mathrm{r}\mathrm{yi}}^{\mathrm{f}}\hfill & \hfill c_{\mathrm{r}\mathrm{zi}}^{\mathrm{f}}\hfill & \hfill c_{\mathrm{r}\mathrm{ui}}^{\mathrm{f}}\hfill & \hfill c_{\mathrm{r}\mathrm{vi}}^{\mathrm{f}}\hfill & \hfill c_{\mathrm{r}\mathrm{wi}}^{\mathrm{f}}\hfill \end{array})$$

(11)

where $c_{\mathrm{r}\mathrm{xi}}^{\mathrm{f}}$ , $c_{\mathrm{r}\mathrm{yi}}^{\mathrm{f}}$ and $c_{\mathrm{r}\mathrm{zi}}^{\mathrm{f}}$ are the three translational compliance (reciprocal of stiffness) coefficients along three Cartesian axes of x, y and z; $c_{\mathrm{r}\mathrm{ui}}^{\mathrm{f}}$ , $c_{\mathrm{r}\mathrm{vi}}^{\mathrm{f}}$ and $c_{\mathrm{r}\mathrm{wi}}^{\mathrm{f}}$ are the three rotational compliance about three Cartesian axes of x, y and z. The coefficients in equation (11) can be determined through FE analysis or semi-analytical fitting.

The closed frame is installed on the fixed base through the guideway assembly, which can be simplified into a virtual lumped spring with compliance in six directions. Apparently, this virtual lumped spring does not provide constraint to the closed frame along the axial direction of the guideway. Therefore, the compliance of the guideway assembly can be formulated as

$${\mathbf{C}}_{\mathrm{r}i}^{\mathrm{g}}=\mathrm{diag}(\begin{array}{cccccc}\hfill c_{\mathrm{r}\mathrm{xi}}^{\mathrm{g}}\hfill & \hfill c_{\mathrm{r}\mathrm{yi}}^{\mathrm{g}}\hfill & \hfill c_{\mathrm{r}\mathrm{zi}}^{\mathrm{g}}\hfill & \hfill c_{\mathrm{r}\mathrm{ui}}^{\mathrm{g}}\hfill & \hfill c_{\mathrm{r}\mathrm{vi}}^{\mathrm{g}}\hfill & \hfill c_{\mathrm{r}\mathrm{wi}}^{\mathrm{g}}\hfill \end{array})$$

(12)

where $c_{\mathrm{r}\mathrm{zi}}^{\mathrm{g}}=0$ . $c_{\mathrm{r}\mathrm{xi}}^{\mathrm{g}}$ , $c_{\mathrm{r}\mathrm{yi}}^{\mathrm{g}}$ and $c_{\mathrm{r}\mathrm{zi}}^{\mathrm{g}}$ are the three translational compliance (reciprocal of stiffness) coefficients along three Cartesian axes of x, y and z; $c_{\mathrm{r}\mathrm{ui}}^{\mathrm{g}}$ , $c_{\mathrm{r}\mathrm{vi}}^{\mathrm{g}}$ and $c_{\mathrm{r}\mathrm{wi}}^{\mathrm{g}}$ are the three rotational compliance about three Cartesian axes of x, y and z. Similar to the closed frame, the coefficients in equation (12) can be determined through FE analysis or semi-analytical fitting.

The lead-screw assembly serially comprises the nut, the lead screw, the front bearing and the rear support bearing, which only provides constraint to the carriage in axial direction. Thus, its compliance can be expressed as

$${\mathbf{C}}_{\mathrm{r}i}^{\mathrm{l}}=\mathrm{diag}(\begin{array}{cccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill c_{\mathrm{r}\mathrm{zi}}^{\mathrm{l}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array})$$

(13)

where $c_{\mathrm{r}\mathrm{zi}}^{\mathrm{l}}$ is the compliance of the lead-screw assembly in the axial direction, and

$$c_{\mathrm{r}\mathrm{zi}}^{\mathrm{l}}=\frac{(l-d_i-l_n)}{\mathrm{EA}}+c_{\mathrm{an}}+c_{\mathrm{af}}+c_{\mathrm{ar}}$$

(14)

where E is the tensile modulus of the lead screw; $A=\pi d_{\mathrm{lead}}^2/4$ , $d_{\mathrm{lead}}$ is the diameter of lead screw; $c_{\mathrm{an}}$ , $c_{\mathrm{af}}$ and $c_{\mathrm{ar}}$ are the compliance coefficients of the nut, front bearing and rear support bearing, respectively.

Substituting equations (10)–(14) into equation (9), one can formulate the compliance of revolute joint assembly as

$${\mathbf{C}}_{\mathrm{r}i}=\mathrm{diag}(\begin{array}{cccccc}\hfill c_{\mathrm{r}\mathrm{xi}}\hfill & \hfill c_{\mathrm{r}\mathrm{yi}}\hfill & \hfill c_{\mathrm{r}\mathrm{zi}}\hfill & \hfill c_{\mathrm{r}\mathrm{ui}}\hfill & \hfill c_{\mathrm{r}\mathrm{vi}}\hfill & \hfill c_{\mathrm{r}\mathrm{wi}}\hfill \end{array})$$

(15)

The elements in the above equation can be calculated in detail as follows: $c_{\mathrm{r}\mathrm{xi}}=c_{\mathrm{r}\mathrm{xi}}^{\mathrm{b}}+c_{\mathrm{r}\mathrm{xi}}^{\mathrm{f}}+c_{\mathrm{r}\mathrm{xi}}^{\mathrm{g}}$ , $c_{\mathrm{r}\mathrm{yi}}=c_{\mathrm{r}\mathrm{yi}}^{\mathrm{b}}+c_{\mathrm{r}\mathrm{yi}}^{\mathrm{f}}+c_{\mathrm{r}\mathrm{yi}}^{\mathrm{g}}$ , $c_{\mathrm{r}\mathrm{zi}}=c_{\mathrm{r}\mathrm{zi}}^{\mathrm{b}}+c_{\mathrm{r}\mathrm{zi}}^{\mathrm{f}}+c_{\mathrm{r}\mathrm{zi}}^{\mathrm{l}}$ , $c_{\mathrm{r}\mathrm{ui}}=0$ , $c_{\mathrm{r}\mathrm{vi}}=c_{\mathrm{r}\mathrm{vi}}^{\mathrm{b}}+c_{\mathrm{r}\mathrm{vi}}^{\mathrm{f}}+c_{\mathrm{r}\mathrm{vi}}^{\mathrm{g}}$ , $c_{\mathrm{r}\mathrm{wi}}=c_{\mathrm{r}\mathrm{wi}}^{\mathrm{b}}+c_{\mathrm{r}\mathrm{wi}}^{\mathrm{f}}+c_{\mathrm{r}\mathrm{wi}}^{\mathrm{g}}$ .

Compliance of spherical joint assembly

The spherical joint connects the limb body to the moving platform and can be simplified as a virtual lumped spring with compliance in three orthogonal translational directions. The compliance matrix of a spherical joint can be expressed as

$${\mathbf{C}}_{\mathrm{s}i}=\mathrm{diag}(\begin{array}{cccccc}\hfill c_{\mathrm{s}\mathrm{xi}}\hfill & \hfill c_{\mathrm{s}\mathrm{yi}}\hfill & \hfill c_{\mathrm{s}\mathrm{yi}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array})$$

(16)

It is worthy to point out that the compliance of a spherical joint is the superposition of the compliances of three revolute joints whose axes are orthogonal to each other and thus varies with joint configurations. The detailed derivation process for the spherical joint stiffness was discussed in Li et al. ¹⁰ and will not be addressed here.

FE formulation of limb body

To simplify the formulation, each limb body is discretized into n − 1 elements with C_i, B_i and A_i being nodes of elements. ε _Ai , ξ _Ai , ε _Bi , ξ _Bi , ε _Ci and ξ _Ci are defined as linear and angular coordinates of nodes A_i, B_i and C_i in the frame B_i-x_iy_iz_i. Similarly, f _Ai , τ _Ai , f _Bi , τ _Bi , f _Ci and τ _Ci are defined as reaction forces and moments at A_i, B_i, and C_i measured in B_i-x_iy_iz_i, respectively. Then, the general coordinates vector u _i and external load vector f _i of the ith limb body can be expressed as

$$\{\begin{array}{c}\hfill {\mathit{u}}_i=(\begin{array}{cccccccc}\hfill {\mathit{\epsilon }}_{\mathrm{Ai}}^{\mathrm{T}}\hfill & \hfill {\mathbf{\xi }}_{\mathrm{Ai}}^{\mathrm{T}}\hfill & \hfill \dots \hfill & \hfill {\mathbf{\epsilon }}_{\mathrm{Bi}}^{\mathrm{T}}\hfill & \hfill {\mathbf{\xi }}_{\mathrm{Bi}}^{\mathrm{T}}\hfill & \hfill \dots \hfill & \hfill {\mathbf{\epsilon }}_{\mathrm{Ci}}^{\mathrm{T}}\hfill & \hfill {\mathbf{\xi }}_{\mathrm{Ci}}^{\mathrm{T}}\hfill \end{array}{)}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{{\mathit{f}}_i=(\begin{array}{cccccccc}\hfill {\mathit{f}}_{\mathrm{Ai}}^{\mathrm{T}}\hfill & \hfill {\mathit{\tau }}_{\mathrm{Ai}}^{\mathrm{T}}\hfill & \hfill \dots \hfill & \hfill {\mathit{f}}_{\mathrm{Bi}}^{\mathrm{T}}\hfill & \hfill {\mathit{\tau }}_{\mathrm{Bi}}^{\mathrm{T}}\hfill & \hfill \dots \hfill & \hfill {\mathit{f}}_{\mathrm{Ci}}^{\mathrm{T}}\hfill & \hfill {\mathit{\tau }}_{\mathrm{Ci}}^{\mathrm{T}}\hfill \end{array}{)}^{\mathrm{T}}}\end{array}$$

(17)

As a result, a set of equilibrium equations of the ith limb in frame B_i-x_iy_iz_i can be formulated with adequate boundary conditions. For convenience, the subscript except the limb body number i (i = 1, 2, 3) is omitted

$${\mathbf{k}}_i{\mathit{u}}_i={\mathit{f}}_i$$

(18)

where k _i is the stiffness matrix of each limb body.

One may define transformation matrices of u _i with respect to linear and angular coordinates of nodes A_i, B_i and C_i as ${\mathbf{N}}_{\mathrm{ci}}^{A1}$ , ${\mathbf{N}}_{\mathrm{ci}}^{A2}$ , ${\mathbf{N}}_{\mathrm{ci}}^{B1}$ , ${\mathbf{N}}_{\mathrm{ci}}^{B2}$ , ${\mathbf{N}}_{\mathrm{ci}}^{C1}$ and ${\mathbf{N}}_{\mathrm{ci}}^{C2}$ . And there exist

$${\mathit{\epsilon }}_{\mathrm{Ai}}={\mathbf{N}}_{\mathrm{ci}}^{A1}{\mathit{u}}_i,{\mathbf{\xi }}_{\mathrm{Ai}}={\mathbf{N}}_{\mathrm{ci}}^{A2}{\mathit{u}}_i,{\mathit{\epsilon }}_{\mathrm{Bi}}={\mathbf{N}}_{\mathrm{ci}}^{B1}{\mathit{u}}_i,{\mathbf{\xi }}_{\mathrm{Bi}}={\mathbf{N}}_{\mathrm{ci}}^{B2}{\mathit{u}}_i,{\mathit{\epsilon }}_{\mathrm{Ci}}={\mathbf{N}}_{\mathrm{ci}}^{C1}{\mathit{u}}_i,{\mathbf{\xi }}_{\mathrm{Ci}}={\mathbf{N}}_{\mathrm{ci}}^{C2}{\mathit{u}}_i$$

(19)

The coordinate transformation can be made to express equation (18) in the reference coordinate system B-xyz as

$${\mathbf{K}}_i{\mathit{U}}_i={\mathit{F}}_i$$

(20)

$${\mathbf{K}}_i={\mathbf{T}}_i{\mathbf{k}}_i{\mathbf{T}}_i^{\mathrm{T}},{\mathit{U}}_i={\mathbf{T}}_i{\mathit{u}}_i,{\mathit{F}}_i={\mathbf{T}}_i{\mathit{f}}_i$$

(21)

where ${\mathbf{T}}_i$ is the transformation matrix of the ith limb body-fixed frame with respect to the global reference and can be expressed as

$${\mathbf{T}}_i=\mathrm{diag}({\mathbf{R}}_i,\dots ,{\mathbf{R}}_i)$$

(22)

Validation of the proposed FE model

In order to verify the accuracy of the above modeling method for the limb body, one individual limb body is taken as an example and its stiffness characteristics are analyzed both with the proposed spatial beam FE model and commercial software ANSYS. To compare this, one end of the limb body (the motor base, C_i) is fixed while a force of 1000 N in the x- and y-directions is applied at the other end (the spherical joint, A_i), respectively. Figure 5(a) and (b) show the deformations of the limb body in the x- and y-directions obtained from ANSYS, respectively. The maximum deflections in the x- and y-directions at the end of the limb body are 0.046 and 0.145 mm while the analytical results from the spatial beam model are 0.047 and 0.141 mm, respectively. The calculation errors are less than 5%. Figure 5(c) and (d) demonstrate the deformations in the x- and y-directions of the limb body along the z-axis obtained from the two methods. It can be found clearly that the results of two models agree very well, indicating that the proposed spatial beam model has a satisfactory accuracy, thus can be used to predict the stiffness characteristics of the limb body.

Figure 5.

Deformations of an individual limb body: (a) deformation nephogram in x-direction, (b) deformation nephogram in y-direction, (c) deformation in x-direction and (d) deformation in y-direction.

Equilibrium equations of the moving platform

The free body diagram of the moving platform is shown in Figure 6. Herein, ${\mathit{F}}_{\mathrm{Ai}}$ is the reaction force vector at the interface between the moving platform and the ith limb body; ${\mathit{F}}_{\mathrm{P}}$ and ${\mathit{\tau }}_{\mathrm{P}}$ are the external forces and moments acting on the moving platform, respectively. And there is

$${\mathit{F}}_{\mathrm{Ai}}={\mathbf{R}}_i{\mathit{f}}_{\mathrm{Ai}}$$

(23)

Figure 6.

Force diagram of the moving platform.

According to Figure 6, the equilibrium equations of the moving platform can be formulated

$$-\sum _{i=1}^3{\mathit{F}}_{\mathrm{Ai}}+{\mathit{F}}_{\mathrm{P}}=\mathbf{0},-\sum _{i=1}^3{\mathit{V}}_{A_i}^B\times {\mathit{F}}_{\mathrm{Ai}}+{\mathit{\tau }}_{\mathrm{P}}=\mathbf{0}$$

(24)

Deformation compatibility conditions

As mentioned above, the moving platform connects with the ith limb assembly through a spherical joint, which can be treated as a virtual lumped spring with equivalent stiffness. The displacement relationship between the platform and the limb can be demonstrated as Figure 7, in which A_iM and A_iL are the interface points associated with the moving platform and RPS limb, respectively. $\nabla {\mathit{A}}_i$ and ${\mathbf{\epsilon }}_{\mathbf{Ai}}$ are the displacements of A_iM and A_iL measured in the limb coordinate system B_i-x_iy_iz_i; k_si is the equivalent stiffness coefficient of the ith spherical joint.

Figure 7.

Displacement relationship between the platform and the limb body.

Observing that the elastic motion of moving platform $U_{\mathrm{P}}={({\epsilon }_{\mathrm{P}}^{\mathrm{T}},{\xi }_{\mathrm{P}}^{\mathrm{T}})}^{\mathrm{T}}$ is caused by the deflection of three flexible limbs and the platform is rigid, one can derive the elastic displacement $\nabla {\mathit{A}}_i$ of A_iM (fixed on the moving platform) as follows

$$\nabla {\mathit{A}}_i={\mathbf{R}}_i^{\mathrm{T}}{\mathbf{D}}^{\mathrm{ri}}{\mathit{U}}_{\mathrm{P}}$$

(25)

where $D^{ri}=[I_{3\times 3}{\widehat{V}}_{A_i}^B]$ ; ${\widehat{V}}_{A_i}^B$ is the reciprocal matrix of vector ${\mathit{V}}_{A_i}^B$ .

Combining equations (19) and (21), the linear coordinate of node A_i can be expressed as

$${\mathit{\epsilon }}_{\mathrm{Ai}}={\mathbf{N}}_{\mathrm{ci}}^{A1}{\mathbf{T}}_i^{\mathrm{T}}{\mathit{U}}_i$$

(26)

As a result, the difference between the elastic displacements of A_iM and A_iL in the limb coordinate system B_i-x_iy_iz_i can be derived

$${\mathit{\epsilon }}_{\mathrm{Ai}}-\nabla {\mathit{A}}_i={\mathbf{N}}_{\mathrm{ci}}^{A1}{\mathbf{T}}_i^{\mathrm{T}}{\mathit{U}}_i-{\mathbf{R}}_i^{\mathrm{T}}{\mathbf{D}}^{\mathrm{ri}}{\mathit{U}}_{\mathrm{P}}$$

(27)

Noting that the spherical joint connects the moving platform with the link at A_i, one can express the reaction forces of the spherical joint as

$${\mathit{f}}_{\mathrm{Ai}}=-{\mathbf{k}}_{\mathrm{s}i}({\mathbf{N}}_{\mathrm{ci}}^{A1}{\mathbf{T}}_i^{\mathrm{T}}{\mathit{U}}_i-{\mathbf{R}}_i^{\mathrm{T}}{\mathbf{D}}^{\mathrm{ri}}{\mathit{U}}_{\mathrm{P}})$$

(28)

Similarly, the reactions at B_i are

$${\mathit{f}}_{\mathrm{Bi}}=-{\mathbf{k}}_{\mathrm{r}1i}{\mathbf{N}}_{\mathrm{ci}}^{B1}{\mathbf{T}}_i^{\mathrm{T}}{\mathit{U}}_i,{\tau }_{\mathrm{Bi}}=-{\mathbf{k}}_{\mathrm{r}2i}{\mathbf{N}}_{\mathrm{ci}}^{B2}{\mathbf{T}}_i^{\mathrm{T}}{\mathit{U}}_i$$

(29)

where ${\mathbf{k}}_{\mathrm{r}1i}=\mathrm{diag}(k_{\mathrm{r}\mathrm{xi}},k_{\mathrm{r}\mathrm{yi}},k_{\mathrm{r}\mathrm{zi}})$ and ${\mathbf{k}}_{\mathrm{r}2i}=\mathrm{diag}(k_{\mathrm{r}\mathrm{ui}},k_{\mathrm{r}\mathrm{vi}},k_{\mathrm{r}\mathrm{wi}})$ are the equivalent linear and angular stiffnesses of a revolute joint in related directions (along and about three axes in limb reference frame B_i-x_iy_iz_i).

Governing equilibrium equations of the system

Inserting equations (28) and (29) into equations (20) and (24), one can write the equilibrium equations of the PKM system as

$$\mathbf{K}\mathit{U}=\mathit{F}$$

(30)

where $\mathbf{K}$ are the global stiffness matrix; U and F are the general coordinate and external load vectors, respectively. And there exist

$$\mathbf{K}=\left(\begin{array}{cccc}\hfill {\mathbf{K}}_{1,1}\hfill & \hfill \hfill & \hfill \hfill & \hfill {\mathbf{K}}_{1,4}\hfill \\ \multicolumn{1}{c}{}& \hfill {\mathbf{K}}_{2,2}\hfill & \hfill \hfill & \hfill {\mathbf{K}}_{2,4}\hfill \\ \multicolumn{1}{c}{}& \hfill \hfill & \hfill {\mathbf{K}}_{3,3}\hfill & \hfill {\mathbf{K}}_{3,4}\hfill \\ \multicolumn{1}{c}{{\mathbf{K}}_{4,1}}& \hfill {\mathbf{K}}_{4,2}\hfill & \hfill {\mathbf{K}}_{4,3}\hfill & \hfill {\mathbf{K}}_{4,4}\hfill \end{array}\right),\mathit{U}=\left(\begin{array}{c}\hfill {\mathit{U}}_1\hfill \\ \multicolumn{1}{c}{{\mathit{U}}_2}\\ \multicolumn{1}{c}{{\mathit{U}}_3}\\ \multicolumn{1}{c}{{\mathit{U}}_4}\end{array}\right),\mathit{F}=\left(\begin{array}{c}\hfill {\mathit{F}}_1\hfill \\ \multicolumn{1}{c}{{\mathit{F}}_2}\\ \multicolumn{1}{c}{{\mathit{F}}_3}\\ \multicolumn{1}{c}{{\mathit{F}}_4}\end{array}\right)$$

(31)

$${\mathit{U}}_4=\left(\begin{array}{c}\hfill {\mathbf{\epsilon }}_{\mathrm{P}}\hfill \\ \multicolumn{1}{c}{{\mathbf{\xi }}_{\mathrm{P}}}\end{array}\right),{\mathit{F}}_4=\left(\begin{array}{c}\hfill {\mathit{F}}_{\mathrm{P}}\hfill \\ \multicolumn{1}{c}{{\mathbf{\tau }}_{\mathrm{P}}}\end{array}\right)$$

(32)

And the global stiffness K can be described in detail as follows

$${\mathbf{K}}_{i,i}=\left(\begin{array}{c}\hfill {\mathbf{R}}_i{\mathbf{k}}_{\mathrm{s}i}{\mathbf{N}}_{\mathrm{ci}}^{A1}{\mathbf{T}}_i^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\mathbf{R}}_i{\mathbf{k}}_{\mathrm{r}1}{\mathbf{N}}_{\mathrm{ci}}^{B1}{\mathbf{T}}_i^{\mathrm{T}}}\\ \multicolumn{1}{c}{{\mathbf{R}}_i{\mathbf{k}}_{\mathrm{r}2}{\mathbf{N}}_{\mathrm{ci}}^{B2}{\mathbf{T}}_i^{\mathrm{T}}}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right)+{\mathbf{K}}_i,{\mathbf{K}}_{4,4}=\left(\begin{array}{c}\hfill \sum _{i=1}^3{\mathbf{R}}_i{\mathbf{k}}_{\mathrm{s}i}{\mathbf{R}}_i^{\mathrm{T}}{\mathbf{D}}^{\mathrm{ri}}\hfill \\ \multicolumn{1}{c}{\sum _{i=1}^3{\mathit{V}}_{A_i}^B\times {\mathbf{R}}_i{\mathbf{k}}_{\mathrm{s}i}{\mathbf{R}}_i^{\mathrm{T}}{\mathbf{D}}^{\mathrm{ri}}}\end{array}\right)$$

(33)

$${\mathbf{K}}_{4,i}=\left(\begin{array}{c}\hfill -{\mathbf{R}}_i{\mathbf{k}}_{\mathrm{s}i}{\mathbf{N}}_{\mathrm{ci}}^{A1}{\mathbf{T}}_i^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{-{\mathit{V}}_{A_i}^B\times {\mathbf{R}}_i{\mathbf{k}}_{\mathrm{s}i}{\mathbf{N}}_{\mathrm{ci}}^{A1}{\mathbf{T}}_i^{\mathrm{T}}}\end{array}\right),{\mathbf{K}}_{i,4}=\left(\begin{array}{c}\hfill -{\mathbf{R}}_i{\mathbf{k}}_{\mathrm{s}i}{\mathbf{R}}_i^{\mathrm{T}}{\mathbf{D}}^{\mathrm{ri}}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right)$$

(34)

Stiffness matrix of the moving platform

In order to evaluate the rigidity of the moving platform, the concept of compliance is adopted, which is mathematically expressed as

$$\mathbf{C}=[{\mathbf{K}}^{-1}{]}_{H\times H}$$

(35)

where H = 18n + 6 is the dimension of the stiffness matrix and n is the number of discrete nodes on each limb body.

With equation (35), the compliance matrix of the platform can be obtained as the last 6 × 6 block matrix in C and is denoted as C_p. Therefore, the stiffness of the platform in the body-fixed frame A-uvw can be expressed as

$${\mathbf{K}}_{\mathrm{p}}={\mathbf{T}}_0^{\mathrm{T}}{{\mathbf{C}}_{\mathrm{p}}}^{-1}{\mathbf{T}}_0={\mathbf{T}}_0^{\mathrm{T}}{\left[{{\mathbf{K}}^{-1}|}_{(H-18n)\times (H-18n)}\right]}^{-1}{\mathbf{T}}_0$$

(36)

where T₀ = diag[R₀R₀] is the transformation matrix of the global reference with respect to the platform body-fixed frame.

Stiffness evaluation of the 3-RPS PKM

The major architecture parameters of a 3-RPS PKM are listed in Table 1. Herein, θ_max denotes the maximum rotation angle of the platform about the u- and v-axes; s represents the stroke of the moving platform; l is the length of the limb body as shown in Figure 3; d_lead is the diameter of the lead screw; d_min denotes the minimum distance between the spherical joint and revolute joint in limb i. The meaning of the other symbols can be referred to the aforementioned context. These architecture parameters are designed to achieve a compromise between global dexterity and rigidity performance over the entire workspace by using the proposed formulation. According to this rule, the workspace of the design is set with motion ranges θ_u = ±40°, θ_v = ±40° and p_z = 540–740 mm.

Table 1.

The architecture parameters of a 3-RPS PKM (unit: mm).

r p	r b	s	d min	l	w 1	w 2	h 1	h 2	θmax (°)	d lead
250	250	200	550	1150	220	200	120	100	±40	35

Table 2 gives the compliance coefficients of three perpendicular axes of a spherical joint in its local frame. In addition, the elastic modulus and shear modulus of the limb body and lead screw are assumed as $E=E\prime =200\mathrm{GPa}$ and $G=G\prime =80\mathrm{GPa}$ , respectively.

Table 2.

The compliance coefficients of S joint in local frames (unit: 10⁻⁵ µm/N).

$c_{su}$	$c_{sv}$	$c_{sw}$	$c_{lu}$	$c_{lv}$	$c_{lw}$	$c_{cu}$	$c_{cv}$	$c_{cw}$
318	300	229	82	370	170	25	55	46

Based on the above parameters of the example system, the following simulations can be carried out.

Stiffness distributions over a given work plane and experimental validation

Figure 8 shows distributions of stiffness values along and about the u-, v- and w-axes over the work plane at p_z = 560 mm. Herein, k₁₁, k₂₂ and k₃₃ denote linear stiffnesses along the u-, v- and w-axes, respectively; k₄₄, k₅₅ and k₆₆ represent angular stiffnesses about the u-, v- and w-axes, respectively; ${\theta }_x=\theta \cos \psi ,{\theta }_y=\theta \sin \psi$ .

Figure 8.

Distributions of stiffnesses when p_z = 560 mm: (a) stiffness along the u-axis, (b) stiffness along the v-axis, (c) stiffness along the w-axis, (d) stiffness about the u-axis, (e) stiffness about the v-axis and (f) stiffness about the w-axis.

It is obvious that the stiffness distributions are strongly position-dependent. For instance, the linear stiffness along the u-axis k₁₁ changes from the minimal value 6.15 × 10⁶ N/m to the maximal value 1.06 × 10⁷ N/m, and the linear stiffness along the v-axis k₂₂ changes from 6.74 × 10⁶ to 1.12 × 10⁷ N/m. The linear principle stiffness along the w-axis k₃₃ varies from 1.48 × 10⁷to 1.14 × 10⁸ N/m. By observing Figure 8(a) and (b), one can find that when the moving platform swings about the x-axis, the linear stiffness along the u-axis k₁₁ decreases while that along the v-axis k₂₂ increases. On the contrary, when the moving platform swings about the y-axis, k₁₁ increases while k₂₂ decreases. Interestingly, there is a common point between k₁₁ and k₂₂ in that their distributions are symmetric about the y-axis. Further observations reveal that distributions of the other four stiffnesses are symmetric, that is, 120° symmetric about the z-axis, which is coincident with the structure feature of the PKM that three RPS limb assemblies are distributed 120° symmetrically. It can also be found that the linear stiffness values along the w-axis k₃₃ are comparatively larger than those of k₁₁ and k₂₂ when the two rotation angles are in a small scale; on the contrary, the angular stiffness values about the w-axis k₆₆ are comparatively smaller than those of k₄₄ and k₅₅ when the two rotation angles are in a small scale. This indicates that the proposed PKM possesses a ‘strong’ rigidity along the tool axis but a ‘weak’ one about the tool axis.

In order to verify the accuracy of the proposed stiffness model, a static stiffness experiment is conducted. Figure 9 illustrates stiffness measurement setups of the 3-RPS PKM. A lifting-jack-like mechanism is used to apply external point forces at the end of moving platform; a force indicator is used to show the magnitude of the applied forces; the deflections subject to external forces at five typical collecting points can be measured by micrometers 1–5. Herein, the 1–4 micrometers are used to eliminate the measurement errors caused by the elasticity of the base.

Figure 9.

Stiffness measurement setups.

Figure 10 illustrates the comparison between the analytical results and experimental results. Figure 10(a) shows the variations of the linear stiffness along the u-axis k₁₁ with respect to p_z. It can be found that both the analytical and the experimental stiffnesses along the u-axis decrease with the increment of p_z. To be specific, the analytical stiffness along the u-axis decreases from 8.35 to 6.48 N/µm with the increment of p_z while the experimental one decreases from 8.34 to 6.43 N/µm with the increment of p_z. Figure 10(b) represents the variations of the linear stiffness along the u-axis k₁₁ with respect to θ_y when p_z = 560 mm, θ_x = 0°. It is obvious that both the analytical and the experimental stiffnesses along the u-axis increase with the increment of θ_y. For instance, the analytical stiffness along the u-axis increases from 8.15 to 9.76 N/µm with the increment of θ_y while the experimental one increases from 7.88 to 10.68 N/um with the increment of θ_y. Further calculation reveals that all the calculation errors between the analytical simulations and experimental tests are less than 10%. Therefore, it can be concluded that the results of the analytical method and experimental tests agree very well, indicating that the proposed stiffness model has a satisfactory accuracy, thus can be used to predict the stiffness characteristics of the 3-RPS PKM.

Figure 10.

Comparison between analytical results and experimental results: (a) stiffness along the u-axis when θ_x = θ_y = 0° and (b) stiffness along the u-axis when p_z = 560 mm, θ_x = 0°.

Stiffness interpretation through eigenscrew decomposition

It can be found from equation (31) that the global stiffness matrix is coupled, which makes it hard to evaluate the rigidity of the moving platform. Therefore, in this subsection, the stiffness behavior of the 3-RPS PKM module is investigated through the eigenscrew decomposition of the stiffness matrix.

The eigenscrew problem formulated in ray coordinates is described as

$${\mathbf{K}}_{\mathrm{p}}\hat{\mathbf{\Delta }}\mathit{\varsigma }=\mathit{\lambda }\mathbf{\varsigma },\hat{\mathbf{\Delta }}=\left[\begin{array}{cc}\hfill \mathbf{0}\hfill & \hfill \mathbf{I}\hfill \\ \multicolumn{1}{c}{\mathbf{I}}& \hfill \mathbf{0}\hfill \end{array}\right]$$

(37)

where the eigenvalue λ is defined as the eigenstiffness of the stiffness matrix K_p; the screw ς is defined as its corresponding eigenscrew; $\hat{\Delta }$ denotes the conversion between two types of coordinates.

Based on equation (37), the eigenscrew decomposition of the stiffness matrix can be expressed as

$${\mathbf{K}}_{\mathrm{p}}=\sum _{i=1}^6{k}_i{\mathit{w}}_i{\mathit{w}}_i^{\mathrm{T}}$$

(38)

where w _i is the unit eigenscrew corresponding to eigenstiffness λ_i; k_i is the spring constant. And there are

$${\mathit{w}}_i=\left(\begin{array}{c}\hfill {\mathit{n}}_i\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_i\times {\mathit{n}}_i+h_i{\mathit{n}}_i}\end{array}\right),k_i=\frac{{\lambda }_i}{2h_i},h_i=\frac{1}{2}{\mathit{w}}_i^{\mathrm{T}}\hat{\Delta }{\mathit{w}}_i$$

(39)

where n _i is an unit vector representing the direction of the spring axis; r _i identifies the direction and distance to the line of action; h_i is the pitch of w _i (i = 1–6).

Equation (38) indicates that K_p can be interpreted by a rigid body suspended six screw springs w _i and six spring constants k_i.

Taking the 3-RPS PKM at a configuration where p_z = 560 mm, θ = 0°, ψ = 0° as an example, the eigenvalues and the corresponding eigenscrews of the system can be obtained by solving the eigenvalue problem of equation (37). The results are listed in Table 3. And the physical properties and geometrical entities of the six screw springs can be listed as in Table 4.

Table 3.

The eigenvalues and eigenscrews of the 3-RPS PKM when p_z = 560 mm, θ = 0°, ψ = 0°.

λi (MN)	λ1 = −12.709	λ2 = −6.3543	λ3 = −6.3543	λ4 = 6.3543	λ5 = 6.3543	λ6 = 12.709
${\mathit{w}}_i=\left(\begin{array}{c}\hfill {\mathit{n}}_i\hfill \\ \multicolumn{1}{c}{{\mathit{r}}_i\times {\mathit{n}}_i+h_i{\mathit{n}}_i}\end{array}\right)$	0	0.9942	0.6327	−0.6330	0.9942	0
	0	0.1077	−0.7744	0.7742	0.1080	0
	−1	0	0	0	0	−1
	0	−0.5200	−0.6259	−0.0841	0.5954	0
	0	−0.4080	0.2129	0.6558	−0.2874	0
	0.1114	0	0	0	0	−0.1114

Table 4.

The equivalent six screw springs based on eigenscrew decomposition when p_z = 560 mm, θ = 0°, ψ = 0°.

Spring	k (106 N/m)	r	h (m)	N
S1	57.028	[0, 0, 0]T	−0.1114	[0, 0, −1]T
S2	5.6642	[0, 0, −0.3500]T	−0.5609	[0.9942, 0.1077, 0]T
S3	5.6642	[0, 0, −0.3500]T	−0.5609	[0.6327, −0.7744, 0]T
S4	5.6642	[0, 0, −0.3500]T	0.5609	[−0.6330, 0.7742, 0]T
S5	5.6642	[0, 0, −0.3500]T	0.5609	[0.9942, 0.1080, 0]T
S6	57.028	[0, 0, 0]T	0.1114	[0, 0, −1]T

As can be seen from Table 3, there exists duality in the system. For example, the first column and the sixth column are dual to each other, meanwhile, the second column and the fifth column are dual to each other. The same rule can be found in Table 4. For example, there exist a dual pair between the first row and the sixth row.

The above eigenstiffnesses and eigenscrews can be converted into a physical description of screw springs with equivalent spring constants. This can be worked out with a parallel mechanism with six screw springs along the six eigenscrews as shown in Figure 11.

Figure 11.

The physical interpretation of the stiffness of a 3-RPS PKM.

When a force produces a parallel linear deformation, and a rotational deformation about the action line produces a parallel couple, a compliant axis occurs. The existence of a compliant axis can also be judged from the eigenscrew solution. The sufficient and necessary condition for the occurrence of a compliant axis is given by the fact that there are two collinear eigenscrews with eigenstiffnesses of equal magnitude and opposite sign. The direction of the compliant axis is coincident with the direction of two collinear eigenscrews. ²⁰ From Figure 11, it can be seen that there exists a compliant axis along the z-axis. The two collinear eigenscrews are in the direction of [0, 0, −1]^T that is, minus the z-axis direction, with eigenstiffnesses of −1.2709 × 10⁻⁷ and 1.2709 × 10⁻⁷ N, respectively. The existence of the compliant axis indicates that the translation along the z-axis is an independent coordinate of the PKM, which demonstrates that one force along the z-axis only arouses a translational displacement along the z-axis while one moment about the z-axis arouses a rotational displacement about the z-axis.

Stiffness evaluation

Global stiffness prediction

To achieve desirable machining accuracy, a PKM module must possess a high rigidity throughout its workspace. Since the stiffness of the 3-RPS PKM is configuration-dependent, a designer may naturally propose that the minimum stiffness should be larger than a specified value throughout the workspace. In addition, the maximal value and average value of the stiffness throughout the workspace should address attentions for optimal design. Therefore, the minimum and maximum as well as the average spring constants of the proposed stiffness matrix are chosen as stiffness indices in order to have a global view of the stiffness performance of the PKM over the whole workspace.

In order to evaluate the stiffness performance throughout the workspace in a quick manner, a numerical approach is applied. The main concept of the numerical simulation is the partition of the whole workspace into discrete sampling pieces which are parallel to z-plane but with different z Cartesian coordinates and the piece-by-piece calculation of the three stiffness indices through eigenstiffness decomposition.

Figure 12 illustrates the distributions of the minimum, maximum and average stiffnesses of the PKM over the work plane at p_z = 560 mm.

Figure 12.

Distributions of stiffness indices throughout the working plane at p_z = 560 mm (θ_x = θ sin ψ,θ_y = θ cos ψ): (a) minimum stiffness, (b) maximum stiffness and (c) average stiffness.

From Figure 12, it can be easily observed that the three stiffness indices of the 3-RPS PKM are axisymmetric, that is, 120° symmetrical about the axial direction of P joints over the given x–y work plane, which is coincident with the axial symmetry of three RPS limbs in the 3-RPS PKM.

Observations also imply a strong dependency of stiffness characteristics of the PKM on the mechanism’s configurations. As can be seen, the minimum of spring constants k_min varies from the lowest value of 4.259 × 10⁶ N/m to the highest value of 5.960 × 10⁶ N/m; the maximum of spring constants k_max changes from 5.663 × 10⁷to 5.745 × 10⁷ N/m and the vary interval of the average spring constant k_ave is [2.279 × 10⁷, 2.353 × 10⁷] N/m.

Further observations reveal that the lowest value of the minimum stiffness occurs at the boundary of the workspace when the moving platform reaches its maximum rotation angle where θ_x = −6.2674° and θ_y = −39.5075°, while that of the maximum stiffness occurs when θ_x = −40° and θ_y = 0°. Different from the maximum and minimum stiffnesses, the lowest value of the average stiffness occurs at the center of the workspace where θ_x = θ_y = 0° and k_ave = 2.279 × 10⁷ N/m.

Since the configuration parameters p_z (or stroke s), the rotation angle θ and precession angle ψ are mainly determined by the working task requirements, the following will analyze the effects of dimensional parameters as well as structural parameters on the stiffness characteristics of the PKM.

Dimensional parameter effect on stiffness indices

Figure 13 shows the variations of the minimum stiffness k_min, maximum stiffness k_max and average stiffness k_ave with respect to the radii of the moving platform and base. Herein, the radius of the moving platform r_p and that of the fixed base r_b vary from 0.20 to 0.35 m, respectively.

Figure 13.

Variations of stiffness indices with respect to r_p and r_b: (a) minimum stiffness, (b) maximum stiffness and (c) average stiffness.

It can be easily observed from Figure 13 that the distributions of k_min, k_max and k_ave are symmetric about the plane of r_p = r_b, which means that the two-dimensional parameters r_p and r_b have the same “intensity” on the stiffness properties of the PKM design. However, the effects of dimensional parameters on the stiffness indices are different about the symmetry plane in that k_max decreases monotonously with the increment of r_p and r_b while k_min and k_ave increase monotonously with the increment of r_p and r_b. With this observation, it might be applaudable to take a strategy of r_p = r_b during the conceptual design stage for this kind of PKM design. Moreover, compared to the minimum and maximum stiffnesses, the effect of the dimensional parameter is very small for the interval of the average stiffness k_ave is [2.2786 × 10⁷, 2.2802 × 10⁷] N/m.

Structural parameter effect on stiffness indices

As aforementioned, the compliance of an individual limb assembly consists of compliances of limb body, the revolute joint and the spherical joint. Thus, the stiffness indices vary with physical parameters of the limb body directly, which affects the stiffness properties of the PKM in turn. Therefore, the following will analyze the effect of cross-sectional parameters of the limb body on the global stiffness indices.

For the convenience of analysis, some physical quantities are defined. For example, define a non-dimensional factor λ_kmin = k_min/k_min0, where k_min0 is the minimum stiffness of the example system when p_z = 560 mm and θ_x = θ_y = 0°. In a similar way, λ_kmax and λ_kave are defined λ_kmax = k_max/k_max0, λ_kave = k_ave/k_ave0. Figure 14 shows the variations of three non-dimensional factors λ_kmin, λ_kmax and λ_kave with respect to δ. Herein, δ denotes the variation of cross sections as shown in Figure 3, that is, the diameters of cross sections I and IV, the length and width of cross sections II and III. It is noted that all the cross sections should be changed with a same variation δ simultaneously.

Figure 14.

Variations of stiffness indices with respect to δ.

From Figure 14, it can be found that the three stiffness indices increase monotonously with the increment of cross sections. For clarity, the intervals of λ_kmin, λ_kmax and λ_kave are [0.9552, 1.0594], [0.9906, 1.0112] and [0.9847, 1.0192]. This is coincident with the common physical explanation, that is, a larger cross section of the limb body helps to improve the rigidity of the limb and thus, in turn, increases the rigidity of the whole PKM system. Comparing the three variation curves, one can find that the limb cross sections have a greater influence on the minimum stiffness k_min of the PKM design for its large variation.

From Figure 14, it seems that an effective way to improve the rigidity of the PKM is to adjust or optimize the dimension of the limb cross sections. However, it is worthy to point out that increasing the limb cross sections will definitely increase the weight and profile of the PKM and may rise other problems such as mechanical interference, which should be considered thoroughly at the design stage.

From the above analysis on Figures 13 and 14, both dimensional parameters and structural parameters have an important effect on the enhancement of the system rigidity. However, the changes in design variables also lead to the changes in dynamics. This may affect the dynamic performance of the PKM when it is applied for high-speed milling, which will be discussed in-depth in our other research article.

Conclusion

A hybrid modeling methodology was proposed to establish an analytical stiffness model for a novel 3-RPS PKM module. With this stiffness model, the rigidity performance of the 3-RPS PKM throughout its workspace was predicted in a quick manner by using a workspace partition algorithm. Based on the proposed model and stiffness analysis, the following conclusions can be drawn: (1) by using a combining method of VJM and FE formulation, the compliances of joints and limbs can be analytically formulated into the stiffness model, making it much more succinct than a numerical stiffness model. (2) The mapping of platform’s stiffness over the working envelope is strongly position-dependent and demonstrates a 120° symmetry about the z-axis, which is coincident with the structure features of the 3-RPS PKM. (3) The calculation errors of stiffness values between the analytical results and experimental tests are all less than 10% for different PKM’s configurations, implying the proposed model can be used to predict the stiffness properties of the 3-RPS PKM throughout its workspace with satisfactory accuracy. (4) In order to have an in-depth understanding of the platform’s rigidity, the stiffness matrix of the platform was extracted from the global stiffness matrix and physically interpreted as a rigid body suspended by six screw springs with equivalent stiffness constants and pitches through eigenscrew decomposition. (5) The parametric analysis indicates that the radii of the platform and the base have the same intensity on the platform’s stiffness while an optimal stiffness can be achieved by adjusting the limb cross sections.

Readers should note that the effects of the gravity, frictions and dampings are not included in this model, which, however, can be easily incorporated into the stiffness model with few efforts. In addition, readers may be aware that the present stiffness model can be further expanded as an elastodynamic model to perform dynamic analysis. These investigations are underway and will be presented elsewhere in the near future.