Development and Application of a Terminal Rigidity Enhancement Methodology for Parallel Robots

Abstract

Parallel robots are recognized as a promising solution for heavy-load and high-efficiency tasks in modern manufacturing systems. The terminal rigidity is a critical performance metric for parallel robots, significantly affecting their payload capacity and machining accuracy. However, enhancing terminal rigidity is still a challenge due to the inherent design conflict between high rigidity and lightweight structures. Aiming at this challenge, this study proposes a practical yet high-ly efficient methodology to enhance terminal rigidity without incurring a weight increase. Firstly, an intensity factor is constructed as a discriminant metric to enable a sensitivity-based classification of components. Subsequently, a rigidity enhancement methodology is developed, which entails executing a step-by-step procedure to improve the stiffness of parallel robots. Next, the methodology is applied to a novel redundantly actuated parallel machining robot to demonstrate its practical utility in engineering design. The result shows that the proposed methodology achieves a 31.3% enhancement in linear terminal rigidity, coupled with a 5.6% decrease in overall weight. Finally, experimental verification is conducted through stiffness tests on a laboratory prototype, demonstrating that the enhanced terminal rigidity meets design expectations.

1. Introduction

The rapid growth of the modern manufacturing industry promotes the development of robotic manipulators towards higher flexibility, higher efficiency, and higher load capacity for implementing heavy-load and high-efficiency tasks [1,2,3,4,5]. Due to the inherent shortcomings of large inertia, low payload capacity, and error accumulation, serial robots struggle to meet these development demands in heavy-load and high-accuracy manufacturing tasks. As an alternative solution, parallel robots were proposed as “plug and play” functional modules for constructing heavy-load robotic manipulators. This is attributed to their advantages of a higher rigidity-to-weight ratio, superior dynamic property and inherently higher precision potential (without error accumulation) compared with serial robots [6,7,8,9,10]. This can be exemplified by the commercial applications of Eco-speed [11,12], Tricept [13,14] and Exechon [15,16] in aeronautics and automotive industries, which demonstrate outstanding flexibility and efficiency in machining large-scale structural components with complex surfaces.

High rigidity is an essential performance requirement for parallel robots engaged in heavy-load yet high-efficiency machining tasks. This is because terminal rigidity directly determines payload capability and anti-deformability of parallel robots, thereby influencing their working efficiency and accuracy. From this perspective, it is important to construct a feasible yet effective methodology to enhance the terminal rigidity of parallel robots in the early design stage. A thorough understanding of the stiffness properties of the robotic system is an indispensable prerequisite. This entails addressing two critical issues: (1) how to characterize the component compliance and the terminal rigidity; (2) how to reveal the effect of component compliance on terminal rigidity.

To address the primary issue of terminal rigidity characterization, an efficient stiffness model and a physically intuitive index framework should be constructed. The former is to reveal the transmission relationship from component rigidity to terminal rigidity, thereby yielding a stiffness matrix. The latter is to quantitatively measure the terminal rigidity of parallel robots.

In the aspect of stiffness modeling, the semi-analytical method and the numerical method are two primary ways. The numerical method is characterized by high accuracy but high computational cost. In contrast, the semi-analytical method is computationally efficient but relatively less accurate, owing to the necessary assumptions and simplifications in its formulation. This implies that the semi-analytical method may be more suitable for iterative design and structure optimization, where computational efficiency is critical. The semi-analytical method can be broadly classified into three categories. The first category formulates the relationship between payload and deformation using the principle of virtual work and static equilibrium equations. In like fashion, Liu et al. [17] proposed a unified approach to derive the stiffness matrix of parallel robots. Similar approaches have been employed in recent studies [18,19,20]. The second category uses the substructure synthesis technique to assemble the stiffness matrices of individual components and to derive the overall terminal stiffness matrix. This category has been successfully adopted to establish various stiffness models for many typical parallel robots [21,22,23]. The third category introduces the strain energy method to formulate the relationship between component compliance and terminal rigidity [24]. This approach has proven particularly effective in stiffness modeling of over-constrained parallel robots [25,26,27].

By tracing the existing studies, the formulation of stiffness indices can be broadly classified into two ways. The first way uses the algebraic characteristics of the stiffness/compliance matrix, such as the inverse of the diagonal elements [17,18,19,20,28], the determinant [29], the condition number [30], or the minimum/maximum eigenvalues [31]. However, such indices often suffer from inconsistent physical units and unclear physical meaning arising from a couple of linear and angular stiffness terms. Alternatively, an equidimensional stiffness index grounded in the virtual work principle was proposed. This stiffness index is formulated as the inverse of virtual work done by a unit force, which provides a direct measure of terminal rigidity along a specific axis. Its effectiveness has been demonstrated in recent evaluations of some over-constrained parallel robots [18,24,25,27].

To address the second issue of quantitative relationship, a number of investigations have been conducted to analyze the influence of design parameters on the terminal rigidity of parallel robots. For example, Sun et al. [26] established the mathematical relationship between stiffness indices and structural parameters for a T5 parallel robot. Li et al. [28] revealed the relationship between structural parameters and stiffness properties of a 3RPS parallel robot. Ceccarelli et al. [32] studied the effects of some basic design parameters on the terminal rigidity of a CaPaMan parallel robot. Furthermore, Wang et al. [33] investigated the influence of structural parameters on component rigidity of a 3SPR parallel robot. Fan et al. [34] investigated the effects of link deformations and joint clearances on the stiffness of heavy-load parallel robots. In addition, the effects of redundant limb structure as well as stiffness parameters on terminal rigidity of parallel robots were quantitatively revealed by the authors [18,21,23]. The studies reviewed above offer important information relevant to the rigidity enhancement of parallel robots.

It is worth pointing out that there is a lack of systematic, generalized approaches for enhancing the rigidity of parallel robots. The most intuitive way to enhance the terminal rigidity is to increase the structural dimension of each individual component in engineering. However, this way is critically limited by its heavy reliance on subjective intuition and trial-and-error. This reliance often results in significant increases in overall weight, moment of inertia, required drive power, and collision risk. Consequently, it may ultimately compromise both kinematic and dynamic performance of the robot system. The present study aims to propose a systematic rigidity enhancement methodology for parallel robots, designed to improve stiffness effectively while avoiding an increase in overall weight.

The rest of this paper is organized as follows. Section 2 constructs an index framework for characterizing terminal rigidity and assessing the effect of component compliance thereon. Building on this foundation, a rigidity enhancement methodology is developed that provides a systematic, step-by-step procedure for improving parallel robot stiffness. Section 3 implements the methodology on a novel redundantly actuated parallel machining robot. Section 4 conducts a set of stiffness tests on a hierarchically fabricated laboratory prototype to validate the methodology. Section 5 presents the conclusions of this study.

2. Rigidity Enhancement Methodology

This section constructs a physically intuitive index framework to characterize stiffness and identify sensitive components. This framework then guides the subsequent development of a general rigidity enhancement methodology. This methodology constitutes a systematic strategy to orchestrate an optimal rigidity-to-weight ratio of parallel robots by selectively tailoring the compliance of both sensitive and insensitive components. Consequently, the presented approach offers a viable way to substantially improve terminal rigidity without compromising on the overall weight of parallel robots.

2.1. General Structure of Parallel Robots

For a topological view, a parallel robot consists of a fixed base, a moving platform, and a group of limbs connecting the moving platform to the fixed base. An individual limb is composed of a set of moving components connected by joints. Each joint can be selected as a spherical joint (denoted as “S”), a universal joint (denoted as “U”), a cylindrical joint (denoted as “C”), a revolute joint (denoted as “R”), or a prismatic joint (denoted as “P”). The limbs can be expressed as the arrangement of joints from the fixed base to the moving platform, such as PRS, UPS, UPR, SPR, R(2SS), and so on. For clarity, Figure 1 demonstrates the schematic diagram of parallel robots.

Figure 1. Schematic diagram of parallel robots.

As illustrated in Figure 1, some necessary definitions and coordinate systems are introduced. B_i and A_i represent geometric centers of the two joints attached to the fixed base and the moving platform in the ith limb, respectively. The points O and O₁ are defined as the feature points (such as the geometric center) of polygons B₁B₂…B_i and A₁A₂…A_i (i = 1–n_L), respectively. n_L is the number of limbs. A global coordinate system O-xyz is attached at point O, while an instantaneous coordinate system O₁-xyz is set at point O₁ with its three axes remaining parallel to those of O-xyz. The stiffness model of such a robotic system has been established with a mature method in the existing studies, which yields a semi-analytical formulation of terminal rigidity as shown in Equation (1).

$$\mathit{K}={\mathit{J}}_L{\mathit{K}}_L{\mathit{J}}_L^T, {\mathit{J}}_L=\left[\begin{array}{cccc}{\mathit{J}}_1& {\mathit{J}}_2& \cdots & {\mathit{J}}_{nL}\end{array}\right], {\mathit{K}}_L=\left[\begin{array}{cccc}{\mathit{K}}_{L,1}& & & \\ & {\mathit{K}}_{L,2}& & \\ & & \ddots & \\ & & & {\mathit{K}}_{L,\mathrm{n}L}\end{array}\right]$$

(1)

where K denotes the terminal stiffness matrix of a parallel robot. J_i (i = 1–n_L) is the Jacobian matrix of the ith limb. K_L,i (i = 1–n_L) is the stiffness matrix of the ith limb. The detailed modeling process can be found in the existing literature of [17,18,19,20].

2.2. Index Construction

This subsection constructs a set of stiffness indices for evaluating the terminal rigidity of a parallel robot by transforming the stiffness matrix into a quantifiable metric. Subsequently, an intensity factor is formulated to reveal the effect of component compliance on terminal rigidity.

2.2.1. Stiffness Indices

To address the unit-induced inconsistent physical meaning, a stiffness metric based on the principle of virtual work is formulated to quantify the deformation resistance of parallel robots. Giving the stiffness matrix K of a parallel robot, the stiffness metric k can be formulated as shown in Equation (2).

$$k={\displaystyle \frac{1}{{\mathsf{\$}}_{F0}^{\mathrm{T}}{\mathsf{\$}}_{\epsilon 0}}}={\displaystyle \frac{1}{{\mathsf{\$}}_{F0}^{\mathrm{T}}\mathit{C}{\mathsf{\$}}_{F0}}}, \mathit{C}={\mathit{K}}^{-1}$$

(2)

where $_F0 denotes a unit wrench imposed on the endpoint of the moving platform, while $_ε0 is the corresponding deflection screw. C represents the compliance matrix of the parallel robot. $_F0, $_ε0 and K are all evaluated in the instantaneous coordinate system O₁-xyz.

Physically, a larger value of k indicates a smaller deflection and a better anti-deformability of the parallel robot. The index k may be adopted as a local stiffness index to evaluate the terminal rigidity along/about the unit wrench $_F0 at a given posture. Based on this foundation, six local stiffness indices are constructed to fully disclose the stiffness characteristics of parallel robots as formulated in Equation (3).

$$\left\{\begin{array}{l}{k}_{Lx}={\displaystyle \frac{1}{{\mathsf{\$}}_{F0,1}^{\mathrm{T}}\mathit{C}{\mathsf{\$}}_{F0,1}}},k_{Ly}={\displaystyle \frac{1}{{\mathsf{\$}}_{F0,2}^T\mathit{C}{\mathsf{\$}}_{F0,2}}},k_{Lz}={\displaystyle \frac{1}{{\mathsf{\$}}_{F0,3}^T\mathit{C}{\mathsf{\$}}_{F0,3}}}\\ k_{Rx}={\displaystyle \frac{1}{{\mathsf{\$}}_{F0,4}^T\mathit{C}{\mathsf{\$}}_{F0,4}}},k_{Ry}={\displaystyle \frac{1}{{\mathsf{\$}}_{F0,5}^T\mathit{C}{\mathsf{\$}}_{F0,5}}},k_{Rz}={\displaystyle \frac{1}{{\mathsf{\$}}_{F0,6}^T\mathit{C}{\mathsf{\$}}_{F0,6}}}\end{array}\right.$$

(3)

where k_Lx, k_Ly and k_Lz are three local linear stiffness indices that evaluate the terminal rigidity along the three axes of O₁-xyz, respectively. k_Rx, k_Ry and k_Rz are three angular stiffness indices that evaluate the terminal rigidity about the three axes of O₁-xyz, respectively. $_F0,i (i = 1–6) is a unit wrench along and about the three axes of O₁-xyz.

Based on the above definition, two global stiffness indices k_gL and k_gR are constructed, i.e., the two minimum values of the above local linear and angular stiffness indices throughout a task workspace. Intuitively, the two global indices k_gL and k_gR reflect the worst linear and the worst angular stiffness of parallel robots. By uniformly discretizing the task workspace, k_gL and k_gR can be approximately calculated by Equation (4).

$$\left\{\begin{array}{l}{k}_{gL}\approx \min \{k_{Lx,1},\cdots k_{Lx,nw},k_{Ly,1},\cdots k_{Ly,nw},k_{Lz,1},\cdots k_{Lz,nw}\}\\ k_{gR}\approx \min \{k_{Rx,1},\cdots k_{Rx,nw},k_{Ry,1},\cdots k_{Ry,nw},k_{Rz,1},\cdots k_{Rz,nw}\}\end{array}\right.$$

(4)

where k_Lx,t, k_Ly,t and k_Lz,t (t = 1–nw) denote the three local linear stiffness indices at the tth discretized posture; k_Rx,t, k_Ry,t and k_Rz,t (t = 1–nw) denote the three local angular stiffness indices at the tth discretized posture; nw denotes the number of discretized postures of the task workspace.

2.2.2. Intensity Factor

The stiffness properties of parallel robots are governed by the compliance of their constituent components. Trial-and-error adjustment of partial structure parameters will be computationally inefficient and lack systematic guidance. The identification of sensitive components is a pragmatic strategy to navigate the high-dimensional design space in such robotic systems.

By referring to the definition of the local stiffness index, the equivalent stiffness coefficient k_c is constructed to evaluate the component compliance about a given direction, as shown in Equation (5).

$$k_c={\displaystyle \frac{1}{{\mathsf{\$}}_{F0}^T{\mathit{J}}_{\epsilon }{(k_e)}^{-1}{\mathit{J}}_{\epsilon }^T{\mathsf{\$}}_{F0}}}$$

(5)

where k_e is the (6-n_f)-order interface stiffness matrix of the component that covers only the constrained axes. n_f represents the degree of freedom. J_ε is the constraint Jacobian matrix of the component.

For a linear system, the effect of an independent variable x on the dependent variable f (x) can be simply evaluated by a gain factor as shown in Equation (6).

$$a_x={\displaystyle \frac{f(x+x_0)-f(x)}{x_0}}$$

(6)

For such a nonlinear mechanical system described in Equation (1), the relationship between the stiffness indices and component compliance exhibits inherent nonlinearity. No constant gain factor exists that can uniformly quantify the influence of the independent variable on the dependent variable. By accounting for the interval variation in the independent variable, an intensity factor (defined as the mean gain factor) is constructed to quantitatively reveal the impact of equivalent stiffness coefficients on the two global stiffness indices. The intensity factor ck can be expressed by Equation (7).

$$c_k={\displaystyle \frac{1}{{\displaystyle {\int }_{-n_h}^{n_h}\mathrm{d}h}}}{\displaystyle {\int }_{-n_h}^{n_h}\left(\left|\frac{k_{g,1}-k_{g,0}}{s-k_c}\right|\right)\mathrm{d}s}$$

(7)

where n_h denotes the given interval boundary of the equivalent stiffness coefficient. kc and k_g,0 denote the initial equivalent stiffness coefficient of the component and its corresponding global linear/angular stiffness index of the parallel robot. k_g,1 represents the iterative global stiffness index with the increase in k_c. c_k evaluates the average variations in the global stiffness index k_g,0 with respect to component compliance k_c with an interval of [−n_h, n_h].

For the sake of computational simplification, the interval of equivalent stiffness coefficients is uniformly divided into a set of nodes. Equation (7) can be approximatively calculated by Equation (8).

$$c_k\approx {\displaystyle \frac{1}{t_h}}{\displaystyle \sum _{i=0}^{t_h}\left|\frac{k_{g,1}-k_{g,0}}{(-n_h+i\cdot h)-k_c}\right|}, t_h={\displaystyle \frac{2n_n}{h}} (-n_h+i\cdot h\ne k_c)$$

(8)

where h and t_h denote the increment of each node and the number of nodes. k_g,1 represents an iterative global stiffness index with kc increased by h.

From the physical perspective, the intensity factor c_k quantifies the influence of the equivalent stiffness coefficient kc on the global stiffness index. A larger c_k indicates a greater influence of the stiffness coefficient on the terminal rigidity of parallel robots. By calculating and comparing the intensity factors of all stiffness coefficients, sensitive and insensitive coefficients may be sorted out rapidly. Consequently, structural parameters that significantly influence a sensitive coefficient are identified as sensitive parameters, while the remaining ones are classified as insensitive parameters.

2.3. Procedure of Rigidity Enhancement

The core principle of the rigidity enhancement is to enhance the terminal rigidity of parallel robots while maintaining or reducing their overall weight in the present study. This can be accomplished by upregulating the size of sensitivity-ranked components to boost terminal rigidity, and concurrently downregulating the size of insensitivity components, thereby optimizing the stiffness-to-weight ratio. It is worth noting that dimensional parameters should remain unchanged during the optimization procedure. This is because they affect the kinematic performance of parallel robots, which is typically determined during the dimension design stage.

The main procedures of rigidity enhancement for parallel robots can be summarized in the following steps:

Step 1: Import the dimensional parameters of the parallel robot to carry out its position solution, and determine a target workspace within the researchable workspace.

Step 2: Obtain and import the initial component stiffness matrices by using finite element software, such as ANSYS 14.5.

Step 3: Substitute component stiffness matrices into a pre-established stiffness model to formulate the terminal stiffness matrix of the parallel robot.

Step 4: Calculate the local stiffness index and find out the feature postures with the worst terminal rigidity within the given task workspace. Herein, m_i (i = 1–t) denotes a motion parameter describing the position or posture of the parallel robot, and t is the number of degrees of freedom. The computation algorithm uses a slice-partition search algorithm with increments of Δm₁, Δm₂,…, Δm_t.

Step 5: Judge whether the parallel robot possesses a desirable stiffness without increasing the overall weight, i.e., k_gL ≥ k_gL,limit, k_gR ≥ k_gR,limit, and G₁ < G₀. Herein, k_gL,limit and k_gR,limit are two design thresholds of the global linear and angular stiffness indices, respectively. G₀ and G₁ denote the overall weight of the initial conceptual design and the enhanced design of the parallel robot. If the parallel robot satisfies those conditions, the design process is ceased. Otherwise, the following steps (Step 6–12) will be performed.

Step 6: Characterize the component rigidity as a set of quantized stiffness coefficients.

Step 7: Calculate intensity factors c_k at the feature postures, with which the effect of each stiffness coefficient on the boundary of terminal rigidity is quantitatively evaluated to sort out sensitive and insensitive stiffness coefficients.

Step 8: Identify the sensitive and insensitive structural parameters that affect those sensitive and insensitive stiffness coefficients based on engineering insight.

Step 9: Adjust the structural features of components using the following way: increasing sensitive parameters to enhance their rigidity and decreasing insensitive parameters to reduce their weight.

Step 10: Recalculate and re-import the optimized component stiffness matrices with the aid of finite element software.

Step 11: Calculate the local stiffness indices at feature postures based on a formulated stiffness model to obtain the global stiffness indices. Herein, t_max is the number of feature postures in the initial conceptual design.

Step 12: Repeat Step 4 to Step 10 until the parallel robot satisfies the conditions of k_gL ≥ k_gL,limit, k_gR ≥ k_gR,limit, and G₁ < G₀.

To facilitate understanding and implementation, Figure 2 illustrates the flowchart of rigidity enhancement for a parallel robot.

Figure 2. Flowchart of rigidity enhancement for parallel robots.

3. Application to a Parallel Machining Robot

In this section, the proposed rigidity enhancement methodology is applied to enhance the terminal rigidity of a novel redundantly actuated parallel machining robot.

3.1. Initial Conceptual Design and Kinematic Definition

For clarity, the initial conceptual design and its schematic diagram of a novel redundantly actuated parallel machining robot (RAPMR) named Z4-RAPMR are depicted in Figure 3.

Figure 3. Initial conceptual design and schematic diagram of the Z4-RAPMR.

As shown in Figure 3a, the Z4-RAPMR has a topology of 2PRU-(2PRU)R, which is composed of a fixed base, a dual platform, and four PRU limbs, denoted as Limb 1, Limb 2, Limb 3, and Limb 4. Each PRU limb contains an actuated prismatic joint, a revolute joint, and a universal joint. For the sake of performance symmetry, the four limbs are designed with identical structures and symmetrical arrangements. The dual platform consists of a primary platform, a revolute joint, and a secondary platform. The revolute joint connects the secondary platform to the primary platform. Specifically, the primary platform is connected to the fixed base by Limb 1 and Limb 3, while the secondary platform is connected to the fixed base by Limb 2 and Limb 4.

As shown in Figure 3b, B_i denotes the original point on the central axis of the ith prismatic joint. The quadrangle B₁B₂B₃B₄ is set as a quadrilateral. Its geometric center is defined as O. A_i represents the geometric center of the universal joint in the ith limb. The midpoints of segments A₁A₃ and A₂A₄ are defined as O₁ and O₂, respectively. C_i is the geometric center of the revolute joint in the ith limb. To be specific, the length of A₁A₃ is set as 515.5 mm. The length of B₁B₃ is set as 520 mm. The length of C_iA_i is set as 390 mm. The length of O₁P is set as 267.5 mm.

In addition, some necessary coordinate systems are defined. A global coordinate system O-xyz is set at O, with its x-axis pointing to B₃, y-axis pointing to B₂, and z-axis satisfying the right-hand rule. Two instantaneous coordinate systems, P-x′y′z′ and O₁-x′y′z′, are attached to P and O₁ with their three axes remaining parallel to those of O-xyz. A local coordinate system O₁-u₁v₁w₁ is set at O₁, with its u₁ axis pointing to A₃, its w₁ axis perpendicular to A₁A₂ and A₃A₄, and its v₁ axis satisfying the right-hand rule. A local coordinate system O₂-u₂v₂w₂ is set at O₂, with its v₂ axis pointing to A₂, its w₂ axis parallel to w₁, and its v₁ axis satisfying the right-hand rule.

The Z4-RAPMR possesses two rotational and one translational motion capabilities [35]. The two rotational axes are respectively parallel to the v₁ and u₂ axes and go through point O, while the translational axis is parallel to the four actuated prismatic joints. By using Euler angles to describe the posture of the Z4-RAPMR, its reachable orientation workspace can be predicted and demonstrated in Figure 4 by a slice-partition search algorithm [35]. Herein, ψ and θ denote two ZYX Euler angles; z is a coordinate value of O₁ along the z axis, measured in the frame O-xyz.

Figure 4. Workspace of the Z4-RAPMR.

As shown in Figure 4, the orientation workspace of the Z4-RAPMR approximates a polyhedron, which is symmetric about the work planes of ψ = 0° and θ = 0°. This is consistent with the fact that those four identical PRU limbs are centro-symmetrically arranged. A cylindrical task workspace is selected with z = [−860, −760] mm and θ_max = 45°. Referring to those parallel machining robots utilized for machining structure components [26,36], the target global linear and angular stiffness indices of the Z4-RAPMR are set as k_gL,tgt = 3.5 × 10⁶ and k_gR,tgt = 5.0 × 10⁵, respectively. Considering manufacturing deviation, a safety factor Q is introduced to obtain the thresholds of the global linear and angular stiffness indices as shown in Equation (9). As a case study, the safety factor Q is set as 1.1 in the present study.

$$k_{gL,\mathrm{limit}}=\mathrm{Q}\cdot k_{\mathrm{g}L,\mathrm{tgt}}=3.85\times {10}^6,k_{\mathrm{g}R,\mathrm{limit}}=\mathrm{Q}\cdot k_{\mathrm{g}R,\mathrm{tgt}}=5.5\times {10}^6$$

(9)

3.2. Stiffness Model Formulation

To facilitate derivation, the Z4-RAPMR is decomposed into a set of components as shown in Figure 5.

Figure 5. Structure of decomposed components.

As shown in Figure 5, an individual PRU limb is further decomposed into five components. To be specific, the first component is a prismatic joint with a carriage assembly; the second component is a revolute joint; the third component is a middle link connecting the revolute joint and the universal joint; the fourth and the fifth components are the two rotational elements of the universal joint, respectively. The dual platform is further decomposed into three components. Herein, the first component is the secondary platform; the second component is the revolute joint connecting the primary platform and the secondary platform; the third component is the primary platform installed with a spindle. To evaluate the rigidity of each decomposed component, a body fixed coordinate system O_i,j-x_i,jy_i,jz_i,j is attached to the jth component in the ith limb. Body fixed coordinate systems O_P1-x_P1y_P1z_P1, O_R-x_Ry_Rz_R and O_P2-x_P2y_P2z_P2 are attached to the first component, the second component and the third component of the dual platform, respectively.

To achieve a symmetrical performance, the same type of components is assumed to have identical structure and stiffness properties. Defining that k_e,i,j and m_e,i,j are the stiffness matrix and the mass of the jth component in the ith limb, respectively. These stiffness matrices are all measured in their corresponding body fixed coordinate systems O_i,j-x_i,jy_i,jz_i,j. k_e,P2, k_e,R and k_e,P1 denote the stiffness matrices of the three components of the dual platform, measured in the body fixed frames O_P2-x_P2y_P2z_P2, O_R-x_Ry_Rz_R and O_P1-x_P1y_P1z_P1. m_e,P2, m_e,R and m_e,S denote their corresponding masses, respectively.

For a serial PRU limb, the stiffness matrix can be formulated as shown in Equation (10).

$${\mathit{K}}_{\mathrm{L},i}=({\mathit{J}}_i^{\mathrm{T}}({\displaystyle \sum _{j=1}^n{\mathit{T}}_{i,j}{\mathit{J}}_{\epsilon ,i,j}{(k_{e,i,j})}^{-1}{\mathit{J}}_{\epsilon ,i,j}^{\mathrm{T}}{\mathit{T}}_{i,j}^{\mathrm{T}}}){\mathit{J}}_i{)}^{-1} (i=1–4)$$

(10)

where K_L,i is a third-order stiffness matrix of the ith PRU limb along/about its constraints. T_i,j denotes the adjoint transformation matrix of the body fixed frame O_i,j-x_i,jy_i,jz_i,j with respect to the instantaneous coordinate system O₁-x′y′z′. J_ε,i,j is the constraint Jacobian matrix of the jth component of the ith limb, defined in the body fixed frame O_i,j-x_i,jy_i,jz_i,j. J_i denotes the Jacobian matrix of the ith limb, which can be expressed as shown in Equation (11).

$$\left\{\begin{array}{l}{\mathit{J}}_i=[{\mathsf{\$}}_{wa,\mathrm{L},i},{\mathsf{\$}}_{wc,\mathrm{L},i,1},{\mathsf{\$}}_{wc,\mathrm{L},i,2}]\\ {\mathsf{\$}}_{wa,\mathrm{L},i}=[{\mathit{s}}_{l,i};{\mathit{r}}_{a,i}\times {\mathit{s}}_{l,i}]\\ {\mathsf{\$}}_{wc,\mathrm{L},i,1}=[{\mathit{s}}_{i,2};{\mathit{r}}_{a,i}\times {\mathit{s}}_{i,2}]\\ {\mathsf{\$}}_{wc,\mathrm{L},i,2}=[{\mathbf{0}}_{3\times 1};{\mathit{s}}_{i,3}\times {\mathit{s}}_{i,4}]\end{array}\right. (i=1–4)$$

(11)

where $_wa,L,i and $_wc,L,i,j denote the unit wrench of actuation and the jth unit constraint wrenches of constraint produced by the ith PRU limb, respectively. r_a,i is the position vector of A_i; s_l,i denotes a unit vector from C_i to A_i. r_a,i and s_l,i are measured in the instantaneous coordinate system O₁-x′y′z′.

By referring to Ref. [36], the stiffness matrix K_(2PRU)R of a (2PRU)R compound limb can be given by Equation (12).

$$\begin{array}{l}{\mathit{K}}_{\left(2\mathrm{PRU}\right)\mathrm{R}}=[{\mathit{J}}_{\left(2\mathrm{PRU}\right)\mathrm{R}}^{\mathrm{T}}({\mathit{J}}_{2\mathrm{PRU}}{\mathit{K}}_{2\mathrm{PRU}}^{-1}{\mathit{J}}_{2\mathrm{PRU}}^{\mathrm{T}}+{\mathit{T}}_{\mathrm{P}2}{\mathit{J}}_{\epsilon ,\mathrm{P}2}{\mathit{k}}_{e,\mathrm{P}2}^{-1}{\mathit{J}}_{\epsilon ,\mathrm{P}2}^{\mathrm{T}}{\mathit{T}}_{\mathrm{P}2}^{\mathrm{T}}+{\mathit{T}}_{\mathrm{R}}{\mathit{J}}_{\epsilon ,\mathrm{R}}{\mathit{k}}_{e,\mathrm{R}}^{-1}{\mathit{J}}_{\epsilon ,\mathrm{R}}^{\mathrm{T}}{\mathit{T}}_{\mathrm{R}}^{\mathrm{T}}){\mathit{J}}_{\left(2\mathrm{PRU}\right)\mathrm{R}}{]}^{-1}\\ {\mathit{K}}_{2\mathrm{PRU}}={\mathit{J}}_{2\mathrm{PRU}}^{\mathrm{T}}({\mathit{J}}_2{\mathit{K}}_{\mathrm{L},2}{\mathit{J}}_2^{\mathrm{T}}+{\mathit{J}}_4{\mathit{K}}_{\mathrm{L},4}{\mathit{J}}_4^{\mathrm{T}}){\mathit{J}}_{2\mathrm{PRU}}\end{array}$$

(12)

where T_P2 and T_R are the adjoint transformation matrices of the body fixed frames O_P2-x_P2y_P2z_P2 and O_R-x_Ry_Rz_R with respect to the instantaneous coordinate system O₁-x′y′z′, respectively. J_ε,P2 and J_ε,R denote the constraint Jacobian matrices of the secondary platform and the revolute joint, measured in O_P2-x_P2y_P2z_P2 and O_R-x_Ry_Rz_R, respectively. J_ε,2PRU and J_ε,(2PRU)R are the constraint Jacobian matrices of two PRU limbs and a (2PRU)R compound limb, respectively. Equation (13) gives their expression.

$$\left\{\begin{array}{l}{\mathit{J}}_{2\mathrm{PRU}}=[{\mathsf{\$}}_{wa,\mathrm{L},2},{\mathsf{\$}}_{wa,\mathrm{L},4},{\mathsf{\$}}_{wc,\mathrm{L},i2,1},{\mathsf{\$}}_{wc,\mathrm{L},i2,2}]\\ {\mathit{J}}_{\left(2\mathrm{PRU}\right)\mathrm{R}}=[{\mathsf{\$}}_{wa,\left(2\mathrm{PRU}\right)\mathrm{R},1},{\mathsf{\$}}_{wa,\left(2\mathrm{PRU}\right)\mathrm{R},2},{\mathsf{\$}}_{wc,\left(2\mathrm{PRU}\right)\mathrm{R}}]\\ {\mathsf{\$}}_{wa,\left(2\mathrm{PRU}\right)\mathrm{R},1}=[{\mathit{s}}_{l,2};{({\mathit{r}}_{a,2}\times {\mathit{s}}_{l,2}+Q_2{\mathit{r}}_{a,2}\times {\mathit{s}}_{2,2})}^{\mathrm{T}}]\\ {\mathsf{\$}}_{wa,\left(2\mathrm{PRU}\right)\mathrm{R},2}=[{\mathit{s}}_{l,4};{({\mathit{r}}_{a,4}\times {\mathit{s}}_{l,4}+Q_4{\mathit{r}}_{a,4}\times {\mathit{s}}_{4,2})}^T]\\ {\mathsf{\$}}_{wc,\left(2\mathrm{PRU}\right)\mathrm{R}}=[{\mathit{s}}_{i2,2};{\mathbf{0}}_{3\times 1}]\end{array}\right.\begin{array}{c}, Q_i=-{\displaystyle \frac{({\mathit{r}}_{a,i2}\times {\mathit{s}}_{l,i2}){\mathit{s}}_{\mathrm{R}}}{({\mathit{r}}_{a,i2}\times {\mathit{s}}_{i2,2}){\mathit{s}}_{\mathrm{R}}}}\\ i2=2,4\end{array}$$

(13)

where $_wa,(2PRU)R,i and $_wc,(2PRU)R denote the ith unit wrenches of actuation and the unit wrench of constraint produced by the (2PRU)R compound limb, respectively.

Since the primary platform is supported by Limb 1, Limb 3, and the (2PRU)R compound limb, the stiffness matrix K_PM of a 2PRU-(2PRU)R parallel robot may be formulated as shown in Equation (14).

$${\mathit{K}}_{\mathrm{PM}}={\mathit{J}}_1{\mathit{K}}_{L,1}{\mathit{J}}_1^{\mathrm{T}}+{\mathit{J}}_3{\mathit{K}}_{\mathrm{L},3}{\mathit{J}}_3^{\mathrm{T}}+{\mathit{J}}_{\left(2\mathrm{PRU}\right)\mathrm{R}}{\mathit{K}}_{\left(2\mathrm{PRU}\right)\mathrm{R}}{\mathit{J}}_{\left(2\mathrm{PRU}\right)\mathrm{R}}^{\mathrm{T}}$$

(14)

where K_PM is measured in the instantaneous coordinate system O₁-x′y′z′.

By considering the compliances of the primary platform and the spindle, the stiffness matrix of the overall Z4-RAPMR can be given by Equation (15).

$$\mathit{K}={({\mathit{T}}_{\mathrm{k}}{\mathit{K}}_{\mathrm{RAPM}}^{-1}{\mathit{T}}_{\mathrm{k}}^{\mathrm{T}}+{\mathit{T}}_{\mathrm{P}1}{\mathit{J}}_{\epsilon ,\mathrm{P}1}{\mathit{k}}_{e,\mathrm{P}1}^{-1}{\mathit{J}}_{\epsilon ,\mathrm{P}1}^{\mathrm{T}}{\mathit{T}}_{\mathrm{P}1}^{\mathrm{T}})}^{-1}$$

(15)

where K is measured in the instantaneous coordinate system P-x′y′z′. T_K and T_P1 are the adjoint transformation matrices of the instantaneous coordinate system O₁-x′y′z′ and the body fixed frame O_P1-x_P1y_P1z_P1 with respect to the instantaneous coordinate system P-x′y′z′, respectively. J_ε,P1 is the Jacobian matrix of the primary platform measured in the body fixed frame O_P1-x_P1y_P1z_P1.

The above stiffness model has been presented and verified by Fang [37]. The experimental results demonstrated that the proposed stiffness model possesses a satisfactory computation accuracy, thus can be used for the stiffness prediction as well as evaluation for the Z4-RAPMR.

3.3. Stiffness Intensity Analysis

Predictably, global stiffness indices and intensity factor calculations will take a huge amount of computation and a lot of time. Existing studies [21,22,23,38] have demonstrated that neglecting the off-diagonal elements in the stiffness matrix of components may not significantly affect the overall stiffness computation. The marginal accuracy loss incurred by this diagonal assumption is outweighed by a substantial gain in computational efficiency. This is particularly valuable for global stiffness evaluation and fast iterative analysis of parallel robots, where precise stiffness analysis results are not required. Given this trade-off, each component is simplified into (6-n_f) linear and angular virtual lumped springs with equivalent stiffness coefficients in the present study. Herein, n_f denotes the degree of freedom of the component. Based on the above simplification, the stiffness matrix k_e,i of the ith component can be expressed as a (6-n_f)-order diagonal matrix, as shown in Equation (16).

$${\mathit{k}}_{e,i}=\left[\begin{array}{cc}{\mathit{k}}_i& \\ & {\mathit{k}}_{w,i}\end{array}\right], {\mathit{k}}_i=\left[\begin{array}{ccc}{k}_{x,i}& & \\ & k_{y,i}& \\ & & k_{z,i}\end{array}\right], {\mathit{k}}_{w,i}=\left[\begin{array}{ccc}{k}_{w1,i}& & \\ & \ddots & \\ & & k_{w(3-nf),i}\end{array}\right]$$

(16)

where k_x,i, k_y,i and k_z,i denote the equivalent linear stiffness coefficients of the ith component along the axes of its body fixed coordinate system; k_w1,i and k_w(3−nf),i denote the equivalent angular stiffness coefficients of the ith component about the constraint axes of its body fixed coordinate system. As demonstrated in Figure 5, the Z4-RAPMR has nine components with 49 equivalent stiffness coefficients. Those stiffness coefficients can be calculated by Equation (5). By using the proposed algorithm, as shown in Figure 2, the two global stiffness indices can be obtained as k_gL = 3.063 × 10⁶ and k_gR = 4.989 × 10⁵, respectively. The worst terminal rigidity of the Z4-RAPMR occurs at the boundaries of the task workspace, i.e., the postures of (ψ = ±45°, θ = 0°, z = −860 mm) and (ψ = 0°, θ = ±45°, z = −860 mm).

By adopting the constructed intensity factor as shown in Equation (7), the effect of those 49 equivalent stiffness coefficients on global stiffness indices can be evaluated quantitatively. By setting each equivalent stiffness coefficient k_e to vary within an interval of [(1–10%)k_e, (1 + 10%)k_e], the intensity factors can be calculated, as depicted in Figure 6. Herein, k_x,i, k_y,i and k_z,i denote three linear equivalent stiffness coefficients, while k_w1,i, k_w1,i and k_w3,i denote three angular equivalent stiffness coefficients of the ith component. These coefficients are measured in the body fixed frame.

Figure 6. Intensity factors calculating results for 49 equivalent stiffness coefficients. (a) Intensity factors with respect to global linear stiffness index. (b) Intensity factors with respect to global angular stiffness index.

As shown in Figure 6, these equivalent stiffness coefficients k_w3,1, k_w1,2, k_w2,2, k_w1,3, k_w2,3, k_w2,4, k_w1,6 and k_w2,6 have relatively larger intensity factors. This indicates that these eight equivalent stiffness coefficients and their corresponding component compliance have significant impacts on the terminal rigidity of the Z4-RAPMR, which are identified as sensitive stiffness coefficients. The remaining 41 stiffness coefficients will be identified as insensitive stiffness coefficients.

3.4. Rigidity Enhancement and Discussion

According to the intensity factor analysis results, the structural parameter optimization of the Z4-RAPMR is carried out at the component level as recorded in this subsection. Herein, the first component of the PRU limb and the second component of the dual platform are two standard components. The structural parameter optimization is implemented by following the underlying idea: increasing the sensitive parameters that exert a significant influence on the eight sensitive stiffness coefficients for high rigidity; decreasing the insensitive parameters that drastically increase mass but do not significantly affect terminal rigidity. For clarity, Figure 7 illustrates the optimization results of the Z4-RAPMR.

Figure 7. Components optimizing for the Z4-RAPMR.

As shown in Figure 7a, structural parameters D_R1 and D_R2 are increased from 114 mm and 28 mm to 124 mm and 30 mm in the PRU limb, respectively. These adjustments are to enhance the angular stiffness of the second component about the axes of x_i,2 and y_i,2 (i = 1–4). As shown in Figure 7b, the third component of the PRU limb has great mass. Its insensitive structural parameters D_L1 and D_L3 are increased from 84 mm and 42 mm to 94 mm and 46 mm, respectively. These adjustments are to enhance the angular stiffness of the 3rd component about the axes of x_i,3 and y_i,3. In addition, structural parameter D_L2 is increased from 2 mm to 12 mm for a mass reduction. As shown in Figure 7c, structural parameters D_U1 and D_U2 are increased from 42 mm and 40 mm to 46 mm and 45 mm in the PRU limb, respectively. These adjustments are to enhance the angular stiffness of the fourth component about the axis of y_i,4. As shown in Figure 7d, the structural parameter D_UZ1/D_UZ2 is increased from 42 mm to 46 mm in the PRU limb. The adjustment is to enhance the angular stiffness of the fifth component about the axes of x_i,5 and y_i,5. As shown in Figure 7e,f, the first and the third components of the dual platform have greater masses. Their insensitive structural parameters D_M1 and D_M2 are decreased from 45 mm and 45 mm to 36 mm and 38 mm, respectively. These adjustments are to reduce the mass of the dual platform.

Table 1 demonstrates the comparison between the initial conceptual design and the enhanced design of the Z4-RAPMR.

Table 1. Comparison between the initial and the enhanced scheme of the Z4-RAPMR.

	Overall Weight/kg	kgL	kgR
Initial scheme	328.49	3.063 × 106	4.989 × 105
Enhanced scheme	310.13	4.023 × 106	5.786 × 105
Relative deviation	5.6%	31.3%	16%

As shown in Table 1, after using the proposed rigidity enhancement methodology, the overall weight of the Z4-RAPMR is reduced from 328.49 kg to 310.13 kg with a reduction rate of 5.6%. Its linear global stiffness index k_gL is increased from 3.063 × 10⁶ to 4.023 × 10⁶ with a growth rate of 31.3%. Its angular global stiffness index k_gR is increased from 4.989 × 10⁵ to 5.786 × 10⁵ with a growth rate of 16%. This result supports that the proposed rigidity enhancement methodology is effective in improving the stiffness of parallel robots without increasing the overall weight.

To further disclose the performance advantage of the enhanced Z4-RAPMR, a comparison study on the local stiffness properties between the initial scheme and the enhanced scheme is carried out. Without loss of generality, the distributions of six local stiffness indices over the work-plane of z = −860 mm are analyzed and shown in Figure 8.

Figure 8. Distributions of six local stiffness indices of the Z4-RAPMR.

As shown in Figure 8, the terminal rigidity of the Z4-RAPMR is strongly position-dependent. The distributions of its six local stiffness indices are simultaneously symmetric with respect to the work planes of ψ = 0° and θ = 0°, respectively. This is consistent with the structural features that four PRU limbs are designed with identical structures and arranged centro-symmetrically. Moreover, these six local stiffness indices of the enhanced scheme are significantly larger than those of the initial scheme throughout the whole work plane. This manifests that the proposed rigidity enhancement methodology is effective in improving the stiffness of parallel robots.

In addition, the proposed rigidity enhancement methodology can be extended to other parallel robots by simply replacing the stiffness model, which is associated with the specific topology and the structural characteristics.

4. Prototype Fabrication and Experimental Validation

4.1. Prototype Fabrication

Based on the aforementioned enhanced scheme, a laboratory prototype of the Z4-RAPMR is hierarchically fabricated, as shown in Figure 9.

Figure 9. Fabrication of a laboratory prototype.

As shown in Figure 9, the fabricated prototype is constructed by assembling four PRU limbs and a dual platform to the fixed base. Herein, an individual PRU limb consists of a prismatic joint and a limb assembly. The prismatic joint assembly adopts a commercial linear motion module, which is actuated by a servo motor. The limb assembly contains a revolute joint and a universal joint. The assembling process can be divided into the following three steps: (1) Assemble four prismatic joints to the fixed base symmetrically and parallelly. (2) Connect a limb assembly to the prismatic joint, in which the revolute joint is perpendicular to the movement axis. (3) Install the dual platform to the four limb assemblies by universal joints. Actuated by the four PRU limbs, the dual platform will achieve two rotational and one translational motion to adjust the postures of the spindle.

The specifications of the measurement instruments used in the stiffness tests are listed in Table 2.

Table 2. The specifications of the measurement instruments used in the stiffness tests.

Instrument	Manufacturer	Parameter	Value
Pressure Sensor	Guangzhou SimBaTouch Electronic Technology Co., Ltd. (Guangzhou, China)	Model	SBT641-500N
		Range (N)	0–500
		Measurement Accuracy	0.3%
Pressure Acquisition Module	Guangzhou SimBaTouch Electronic Technology Co., Ltd. (Guangzhou, China)	Model	SBT951
		Display Accuracy (N)	0.1 N
Dial Indicator	Dongguan Quick Measuring Tools and Instruments Co., Ltd. (Dongguan, China)	Resolution (mm)	0.001
		Range (mm)	0–10

4.2. Stiffness Test and Validation

To verify the effectiveness of the proposed rigidity enhancement methodology, a set of stiffness tests is carried out on the fabricated prototype. The underlying idea is to make a comparison study between the experimental stiffness indices and the target global stiffness index. This idea can be implemented through the following three steps.

Step 1: Construct an experiment system for stiffness testing and select a set of reasonable measurement configurations to demonstrate the worst terminal rigidity of the Z4-RAPMR.

Step 2: Measure and obtain the load-deflection data by using the established experiment system, based on which the stiffness indices of the laboratory prototype are calculated.

Step 3: Compare the experimental stiffness indices with the target stiffness index to verify the effectiveness of the proposed rigidity enhancement methodology.

For clarity, Figure 10 depicts the full experimental setup of stiffness testing.

Figure 10. Stiffness testing set-up.

As depicted in Figure 10, the screw jack is used to apply force to the Z4-RAPMR, while the pressure sensor is adopted to measure its magnitude. Moreover, a dial indicator is installed in a bracket to measure the deflection under pressure. The proposed rigidity enhancement methodology adopts global stiffness indices as its design objective, thereby ensuring the worst-case terminal rigidity of parallel robots. Consequently, the stiffness testing in the present study will focus specifically on those configurations where global stiffness indices are observed. As illustrated in Figure 8, the global linear stiffness index is located at the extreme postures of either (ψ = ±45°, θ = 0°, z = −860 mm) or (ψ = 0°, θ = ±45°, z = −860 mm). For this purpose, these two extreme configurations with maximum θ and ψ were selected to validate the worst-case scenario as depicted in Figure 11.

Figure 11. Stiffness testing for two extreme configurations.

As shown in Figure 11, the load-deflection data of the Z4-RAPMR can be obtained by applying a set of forces to the endpoint. To achieve satisfactory testing accuracy, a preloading force is applied to eliminate joint clearance during the experimental testing. The preloading force and its induced deflection are not accounted for in the present experiments. At each experimental configuration, the applied force was increased gradually, during which its magnitude and the terminal deflection of the parallel robot were measured and recorded. The work W done by the applied force can be expressed as Equation (17).

$$W=\mathit{F}\mathit{\delta }={\mathit{F}}^{\mathrm{T}}\mathit{C}\mathit{F}={\epsilon }^2{\mathsf{\$}}_F^{\mathrm{T}}\mathit{C}{\mathsf{\$}}_F=\epsilon \delta , \mathit{C}={\mathit{K}}^{-1}$$

(17)

where F and δ are the applied force and its induced deflection, respectively. ε and $_F denote the magnitude and the unit wrench of the applied force, respectively. δ is the deflection along the applied force. Substituting Equation (1) into Equation (17), the local stiffness index may be calculated as shown in Equation (18).

$${\epsilon }^2{\mathsf{\$}}_F^{\mathrm{T}}\mathit{C}{\mathsf{\$}}_F={\displaystyle \frac{{\epsilon }^2}{k}}=\epsilon \delta \to k={\displaystyle \frac{\epsilon }{\delta }}$$

(18)

Following the experimental tests, the experimental stiffness indices were compared with the target global stiffness index as illustrated in Figure 12. Herein, “Predicted deflection” is calculated by the established stiffness model, while “Tested deflection” is obtained by the dial indicator directly. “Absolute deviation” denotes the absolute deviation between “Tested deflection” and “Predicted deflection”. “Experimental stiffness index” is calculated by Equation (18).

Figure 12. Stiffness testing results for the two extreme configurations.

As can be seen clearly from Figure 12, the experimental stiffness indices show a slight fluctuation with variation in applied force. This may indicate that the fabricated Z4-RAPMR possesses a stable terminal rigidity, and the stiffness testing is reasonable. Such a slight fluctuation may be caused by non-ideal contact and inhomogeneous dampers existing in the joints. Compared with the target global stiffness index k_gL,tgt, the experimental stiffness indices all have larger values. The results support that the proposed rigidity enhancement methodology is effective in improving the terminal rigidity of parallel robots with a satisfying global stiffness index. Additionally, it is worth noting that the terminal deflections obtained from the experimental tests are larger than those predicted by the established stiffness model. Their largest absolute deviation is 5.45 μm with a relative error under 10%. In other words, the established stiffness model may overestimate the stiffness indices of the parallel robot. The main reason lies in the fact that the established stiffness model makes some assumptions and approximations: (1) the components are assumed to be linearly elastic; (2) all manufacturing deviations and joint clearances are neglected. Such stiffness overestimation can be rectified through calibration in actual engineering applications, thereby ensuring more accurate performance predictions. So, it is important to set a reasonable safety factor for the target global stiffness index in the design stage.

5. Conclusions

The main findings of this investigation are summarized as follows:

(1)

A set of local and global stiffness indices is constructed to evaluate the terminal rigidity of parallel robots across their task workspace. Building on this foundation, an intensity factor is subsequently introduced to quantitatively reveal the effect of component compliance on the terminal rigidity of a parallel robot.

(2)

A practical and highly efficient methodology is developed to enhance the terminal rigidity of parallel robots while maintaining or reducing their overall weight. This is attributed to the use of an intensity factor to quickly identify sensitive and insensitive stiffness coefficients, thereby enabling an optimal balance between high rigidity and low weight.

(3)

To demonstrate its practical application in designing high-rigidity parallel robots, the proposed rigidity enhancement methodology is applied to a novel Z4-RAPMR. The implementation leads to 31.3% and 16% increases in global linear and angular stiffness indices, while achieving a 5.6% reduction in overall weight.

(4)

A stiffness testing set-up is developed and employed to conduct a set of terminal rigidity tests on a laboratory prototype. The testing results revealed that the experimental stiffness indices at two extreme configurations surpassed the predefined global stiffness target. This suggests that the enhanced Z4-RAPMR may possess a satisfactory terminal rigidity throughout its target workspace. This successful application verifies that the proposed methodology can effectively enhance the terminal rigidity of parallel robots.

(5)

It is worth noting that the present study does not account for potential stiffness degradation due to non-ideal contacts or inhomogeneous dampers. Future work will therefore focus on investigating the effects of joint clearance, bearing preload, and overconstraint on the terminal rigidity of parallel robots.