Stiffness Evaluation of a Decoupled Cartesian Parallel Manipulator Open Access

The high stiffness-to-mass ratio is often seen as a major advantage of parallel manipulators. In the design stage of a parallel manipulator, one typically evaluates the manipulator's stiffness and subsequently optimizes it along with other important measures, such as mass and workspace, in a multi-objective optimization. The stiffness analysis evaluates the deformation of the manipulator with certain geometry, boundary conditions, and material, due to an applied wrench. The manipulator's stiffness can be expressed as a stiffness matrix or a resulting deformation due to a specified applied wrench. In the former case, the elements of the stiffness matrix indicate the amount of wrench elements required to cause a unit deformation in the constrained degrees-of-freedom (DOF). In the latter case, the amount of deformation due to the specified wrench indicates the stiffness. There are two common purposes for stiffness evaluation of a manipulator. The first purpose, which is the purpose of this work, is to be used for design optimization. This usually aims to optimize the dimension of the manipulator while assuming the use of a certain material. With this goal in mind, an overall optimization of the manipulator geometry typically starts with the optimization of the links' lengths. Once the optimum links' lengths are obtained, other geometric parameters, such as the cross-sectional size and the detailed links' topology, can be subsequently optimized. The second purpose of a stiffness evaluation is to quantify the absolute deformation of the manipulator given a certain wrench. When this is the goal, it is important to include all the stiffness components or at least all the significant stiffness components. Furthermore, the detailed geometry and material of the manipulator's parts should be considered when the absolute value of the manipulator's deformation is the concern. In the following section, several stiffness evaluation methods will be reviewed to discuss their advantages and disadvantages, and the methods will be used selectively in this article.

Based on the coupling condition, parallel manipulators can be classified into fully coupled, partially decoupled (group decoupled), and fully decoupled manipulators [1]. In fully decoupled manipulators, each end-effector (EE) coordinate depends on only one active joint coordinate independently [2]. In other words, the task space coordinates and the joint space coordinates have a one-to-one correspondence relationship. While the stiffness of a decoupled parallel manipulator can be analyzed in a similar manner to that of general parallel manipulators, the authors cannot find a work discussing a method to evaluate specifically the stiffness of a decoupled parallel manipulator except Ref. [3]. Nevertheless, the stiffness evaluation in that work has the following limitations:

• The work only performed the stiffness evaluation in one Cartesian direction, and it did not evaluate the manipulator stiffness subject to a combined force, i.e., a force consisting of non-zero components working in all the Cartesian directions.

• The work did not investigate the stiffness of the manipulator due to an applied moment.

• The work did not explicitly present a systematic method to evaluate the stiffness of the entire manipulator subject to general force and moment applied to the end-effector.

• The work did not study how the variation of the manipulator kinematic parameters affects the manipulator stiffness.

In this work, we overcome the aforementioned limitations by proposing a systematic approach and performing parametric studies to evaluate the stiffness of a decoupled parallel manipulator. This work was driven by our need to evaluate the stiffness of a 3DOF decoupled Cartesian parallel manipulator [4,5] composed of three serial RRR links in a quick and intuitive manner. We will start by evaluating the stiffness of an individual limb with two links connected by R joints with parallel joints' axes. The study on the individual two-link limb focuses on the behavior of the limb before it is assembled into the entire manipulator. Subsequently, we will evaluate the stiffness of the entire 3DOF decoupled Cartesian parallel manipulator using a systematic approach we propose. The idea of the approach is to break down the stiffness evaluation into multiple separate analyses corresponding to elementary Cartesian wrench components resolved from a general wrench applied to the manipulator's end-effector. The separate analyses are performed based on the decoupled behavior of the Cartesian parallel manipulator. The proposed method is simpler and more intuitive than the standard stiffness evaluation method, which requires an assembly procedure. However, it comprises multiple analyses and only applies to decoupled Cartesian parallel manipulators. Using the approach we propose, as will be shown in this article, the stiffness of the entire manipulator can be evaluated at the individual limbs in a simple manner and subsequently completed by superposing the results to obtain the stiffness of the entire manipulator. Using the proposed approach, a simpler and more intuitive stiffness evaluation can be performed in the manipulator design stage. This study complements the previous work [3] by the following: (1) proposing a systematic method to estimate the stiffness of the manipulator subject to combined force and moment, (2) comparing the result of the proposed approach with the standard approach that is based on an assembled manipulator model. Furthermore, this study also aims to investigate the influence of the manipulator's kinematic parameters on its stiffness. Since the aim of the stiffness evaluation is topology and dimensional optimization, the compliance of the manipulator in this work is assumed to be given by the compliance of its links. The manipulator's joints are assumed to be rigid.

The remainder of this article is organized as follows. Section 2 briefly reviews the methods used to evaluate the stiffness of a manipulator. Section 3 presents the analysis setup used in this work, including the model boundary conditions, material properties, and geometrical dimensions of the analyzed mechanism. Section 4 evaluates the out-of-plane stiffness of the individual two-link limb. The term “plane” here is defined as the plane in which the links of the limb perform their motion. After the discussion of the individual limb stiffness, Sec. 5 presents our simplified approach to evaluate the stiffness of an entire decoupled Cartesian parallel manipulator without a need to perform assembly. Finally, Sec. 6 concludes the article. Throughout the article, the revolute and prismatic joints are indicated by R and P, respectively. An underlined joint indicates that the joint is actuated.

The stiffness modeling of a manipulator can be broadly classified into lumped and distributed modeling methods. The most commonly used lumped model is the Jacobian-based virtual joint model (VJM). Using this method, it is quite common to select one or more stiffness components, such as the actuator stiffness [6–11], the axial stiffness of the links [8–12], the bending stiffness of the links [7,9,10], and the torsional stiffness [11] of the links. To incorporate the stiffness in all translational and rotational directions, Pashkevich et al. [13] proposed a VJM method that uses 6DOF springs to represent the stiffness of the manipulator's links. A VJM method based on the overall Jacobian [14] that consists of the actuation and constraint Jacobians defines stiffness as the relationship between the applied wrench and the displacements of the actuations and the constraints [11,15]. While the determination of the actuators' stiffness is quite straightforward, the determination of the constraints' stiffness is not always straightforward. Hoevenaars et al. [16] proposed a VJM method based on the generalized Jacobian [17] that is broader than the overall Jacobian. Compared with the VJM method proposed by Pashkevich et al. [13], this method results in a smaller dimension, i.e., the size of matrices. However, this advantage is most likely not quite significant, particularly with the current computing hardware technology, as any VJM method does not have a high dimension.

The most commonly used distributed models are the matrix structural analysis (MSA) and finite element analysis (FEA). The FEA is convenient to accommodate complex geometries with high accuracy by using suitable element types and meshing strategies. An experimental investigation by Pinto et al. [3] shows that the FEA model is only different by 0–2% from the experiment. However, the stiffness evaluation using the FEA should be performed for every manipulator's posture since an FEA user typically does not have the parametrical expressions of the finite element model of the manipulator. The MSA, sometimes called the direct stiffness method, can be seen as a special case of the FEA as it is basically FEA performed through matrix operations that are conveniently used for one-dimensional finite elements such as bars and beams. Since the MSA is only applied to simple one-dimensional finite elements, it is quite feasible to code the parametrical expressions of the model. Accordingly, the model can be used across the postures of the modeled manipulator without a need to re-setup for every posture. Klimchik et al. [18] proposed a new formulation of the MSA in which they introduced boundary condition equations (matrices) that eliminate the necessity to remove the rows and columns corresponding to the boundary conditions from the stiffness matrix as in the conventional MSA [19].

Pashkevich et al. [13] and Klimchik et al. [20] proposed a hybrid approach that combines VJM with the FEA through stiffness matrix identification using the FEA. This is achieved by performing virtual experiments in an fea software. Another hybrid approach is performing a model order reduction to an FEA model. Taghvaeipour et al. [21] obtained the stiffness matrices of complex anisotropic links through substructuring to obtain reduced-order stiffness matrices. Furthermore, Cammarata [22] employed several model order reduction methods, namely Guyan, Craig–Bampton, and Improved Reduced System (IRS) methods, to reduce the order of a finite element model of a multibody elastodynamic system.

Stiffness indices are typically used as a handy scalar indicator of the manipulator stiffness. The VJM provides a 6 × 6 stiffness Cartesian matrix (CSM). The stiffness magnitude is commonly measured by using the trace and the weighted trace of the CSM, the minimum, maximum, and average eigenvalues of the CSM, the mean value of the CSM, and sometimes the determinant of the CSM. The stiffness uniformity is typically measured by using the condition number of the CSM. Despite the convenience of extracting a stiffness index from the CSM, the stiffness index suffers from the non-uniformity problem when a manipulator mixes translational and rotational degrees-of-freedom. There are typically two strategies to deal with such a problem. The first strategy is separating the evaluation between the translational and rotational stiffness components. The second strategy is homogenizing the stiffness matrix. Some global indices such as the global compliance index [7] and the global stiffness index (GSI) [23] are commonly used to evaluate a manipulator's stiffness across the workspace.

The distributed stiffness models such as MSA and FEA typically do not provide a 6 × 6 Cartesian stiffness matrix. Using these models, the computed deformation of the end-effector is typically used as the base of a stiffness index. A global stiffness index is typically obtained by averaging the deformations across the workspace. To have an alternative stiffness index based on a distributed stiffness model, Raoofian et al. [24] proposed an enhanced MSA model, verified it with an FEA model, and proposed a method to obtain a 6 × 6 CSM from the enhanced MSA model. Subsequently, they proposed a stiffness index based on the obtained 6 × 6 CSM.

Figure 1 shows the two-link limb to be evaluated in this work. This two-link limb also serves as the main mechanism of the decoupled Cartesian parallel manipulator discussed at the end of this article. Point O is the fixed position of the proximal R joint. The proximal link makes an angle α₁ with respect to the X-axis. This angle is called the limb angle across this article. In practice, the virtual segment connecting point O and the end-effector can be oriented at any angle. For convenience, this virtual segment will be illustrated as a horizontal line coincident with the X-axis, implying that the end-effector keeps its rigid body position along the X-axis despite the change of the angle $θ$ and the links' lengths $L1$ and $L2$⁠. Assuming that each R joint is passive, the moment about the joint axis is not constrained; hence, it cannot be modeled as a fixed joint. However, when the two-link limb with passive R joints is subject to only an out-of-plane load, the passive R joints can be modeled as fixed joints. In this work, the two-link limb is only subject to an out-of-plane load. Accordingly, the passive R joints are modeled as fixed joints.

Table 1 shows the material properties and geometrical dimensions of the links used in this work. The links are made from an aluminum alloy with the mentioned properties. Unless stated otherwise, the links of the mechanism have identical lengths, i.e., $L1=L2$⁠.

The EE deformation in this work is evaluated using the analytical, MSA, and FEA methods. We proposed some analytical approximation in this work to evaluate the end-effector deformation. We coded the MSA solver using the Euler–Bernoulli beams with six degrees-of-freedom at each node to model the links. Each link has only nodes at its ends. On the other hand, we obtained the FEA solution using ansys mechanical. The FEA model discretizes each link into 20 beam elements. Figure 3 shows the FEA models in ansys mechanical. Point A is fixed. Point B is the end-effector of the two-link limb. In Fig. 3(a), the out-of-plane force, working in a direction perpendicular to the paper, is applied at point B. In Fig. 3(b), the out-of-plane moment, with its rotation axis lying on the plane and perpendicular to the link, is applied at point B. The parametric evaluation in the ansys fea was performed for several discrete values of the parameters within the range of interest. To evaluate the stiffness across the workspace, we plotted stiffness maps based on the end-effector deformation computed using the MSA method.

Now consider the two-link limb in three special cases, namely fully extended, fully folded, and right-angled configurations, as depicted in Fig. 4. Although the three configurations are typically feasible to achieve in a single limb, some of them, particularly the fully extended and fully folded configurations, can be impractical to achieve in multiple limbs composing a parallel manipulator, due to the kinematic constraints of the manipulator. The out-of-plane deformation at the end-effector due to an out-of-plane force $Fz$ and an out-of-plane moment $My$ in the three special cases, as shown in Table 2, can be derived using the textbook mechanics of materials. The analytical solutions shown in Table 2 are identical to those derived from Eqs. (3) and (4) by setting the angles to values corresponding to the special cases. The table also compares the analytical solution with the MSA and FEA solutions. Since the analytical and MSA solutions are identical, the percentage error shown in the table indicates the difference between either of them and the FEA solution.

Figure 5 shows one-link and two-link limbs with an out-of-plane force, perpendicular to the paper, applied to the tip of the limbs. For simplicity, let us assume a one-link limb. In this simple case, the bending problem is similar to that in the bending of a straight cantilever beam. The smaller the limb length L, the stiffer the limb is, and vice versa. In the case of the two-link limb, the appearance of the limb is not only dictated by the effective length of the limb, L, but also W₁ and W₂. Assuming an identical length of the two links, W₁ = W₂ = W, L and W are dependent on each other if the link length is already given. The smaller L is, the larger W is, and vice versa.

First of all, let us assume that the one-link and two-link limbs have identical link cross section and material. If both the limbs have identical L, we can hypothesize that they do not have an identical stiffness. Furthermore, if two-link limbs have identical L but different W, we also hypothesize that they do not have identical stiffness. In a single two-link limb, we are interested in evaluating its stiffness at its varied posture. In this case, a specific posture is defined by its limb angle, effective length L, and folding depth W, which depend on each other. To study the effects of the parameter values, parametric studies of a two-link limb were performed using analytical analysis based on the analytical, MSA, and ansys fea solutions.

Here, the out-of-plane stiffness of the limb is evaluated, and the angle of the limb is varied. Figure 6 shows the out-of-plane end-effector deformation of the limb due to a separate out-of-plane force of 50 N and an out-of-plane moment of 25 Nm. The figure shows that the out-of-plane stiffness is affected by the limb angle. Furthermore, the data show that the larger the limb angle $α1$⁠, the stiffer the limb in the out-of-plane direction. This is an intuitive behavior since a larger limb angle $α1$ corresponds to a shorter effective length of the limb. The data also show that the analytical and MSA solutions are identical. Both solutions do not have a significant difference from the FEA solution at the fully extended and fully folded configurations. The closer the limb to the right-angled configuration, the larger the difference between the analytical/MSA and FEA solutions. As shown in Table 2, the largest difference is around 10%, occurring at the right-angled configuration.

First, the links' lengths vary while the total length of the two-link limb is maintained constant. The effective length is not maintained to be constant. This evaluation was performed at three representative limb angles, namely 30 deg, 50 deg, and 80 deg. Figure 7 shows the out-of-plane end-effector deformation in this evaluation. The figure shows that the end-effector deformation follows the effective length. The smaller the effective length, the smaller the end-effector deformation, and vice versa.

To exclude the influence of the effective length, another evaluation was performed by varying the link lengths while keeping both the total length and effective length of the limb constant. Accordingly, each pair of $L1$ and $L2$ corresponds to a unique limb angle. This evaluation was performed in the MSA and ansys fea. In this evaluation, an out-of-plane force of 100 N and an out-of-plane moment of 25 Nm are separately applied to the end-effector. Figure 8(a) shows the ansys fea model, whereas Fig. 8(b) shows the limb postures at all the varied link lengths. The out-of-plane end-effector deformation of the limb is shown in Fig. 9. The figure shows that a larger link length ratio results in a smaller out-of-plane end-effector deformation.

After discussing the stiffness of a two-link limb with passive R joints, let us discuss the stiffness of a 3PRRR parallel manipulator, which has three RRR limbs with passive R joints and loads applied perpendicular to the motion plane of the limbs. An actuated P joint is added to the proximal end of each limb, transforming the limb from RRR to PRRR topology. Figure 10(a) shows the 3PRRR parallel manipulator. The underlined letter indicates the actuated joint. It is a 3DOF decoupled Cartesian parallel manipulator. The first link in each limb is typically very short, so its stiffness can be neglected. Accordingly, the manipulator topology can be simplified, as shown in Fig. 10(b). Since all the R joints in each limb are parallel, the limbs' motion planes are always parallel with the Cartesian planes, namely YZ, XZ, and XY planes.

For convenience, let us see the parallel manipulator at one of its postures in a two-dimensional view, as depicted in Fig. 11. For simplicity, let us assume that the manipulator's end-effector is only subject to an external force. In other words, an external moment is not applied to the end-effector. With the P joints locked at their positions, namely $x1$⁠, $x2$⁠, and $x3$⁠, let us apply an external force to the manipulator's end-effector. This force can be resolved into three components, namely $Fx$⁠, $Fy$⁠, and $Fz$⁠, as depicted in Fig. 10.

When an external force in the X-direction, namely $Fx$⁠, is applied, the locked limb 1 prevents the end-effector from a rigid body translation in the X-direction. Accordingly, limb 1 is suffering from a transversal bending in the X-direction due to the force $Fx$⁠. This leads to a deformation in the X-direction, namely $dx1$ (with the superscript 1 indicating the limb and the subscript x indicating the deformation direction). Meanwhile, limbs 2 and 3 cannot resist the deformation of limb 1 in the X-direction since all the R joints in both limbs are passive. This is a physical consequence of the decoupled characteristics of the manipulator. The out-of-plane deformation of a limb is only resisted by the structure of that particular limb. A similar behavior applies to limbs 2 and 3 when forces $Fy$ and $Fz$ are applied, respectively. This behavior is summarized in Table 3.

Consequently, the end-effector's deformations $dx1$⁠, $dy2$⁠, and $dz3$ are only resisted by the individual limbs 1, 2, and 3, respectively. More specifically, the resistance comes from the structural constraint of the individual limb that behaves as a cantilever subject to bending. When an external force applied to the end-effector has a diagonal direction, i.e., not purely in X, Y, or Z direction, then two or more limbs are subject to bending. By resolving the external force into its Cartesian components, a separate analysis can be performed for each loaded limb.

On the other hand, rigid body rotations about the X, Y, and Z axes are constrained collectively by the three limbs. These rotational constraints are also handled by the structure of the limbs. Table 4 summarizes the constrained rotations in each limb. Since the constraints of the end-effector are the sum of the constraints of the limbs, it is easy to deduce from Table 4 that the end-effector is constrained by rotation about the X, Y, and Z axes. Accordingly, the manipulator has only 3T degrees-of-freedom.

It is worth mentioning that the rotational constraints exist at all times, regardless of whether the actuators are locked or not. In contrast, the translational constraints theoretically apply only when the corresponding actuators are locked. Nevertheless, in practice, if the translational actuation is implemented using lead screws, the translational constraints at all times can be achieved due to the translational resistance of the lead screw mechanism, regardless of whether the actuators are locked or not.

Consider the 3PRRR manipulator with a moment applied to its end-effector. The moment can be resolved into three components about the X-, Y-, and Z-axis, namely $Mx$⁠, $My$⁠, and $Mz$⁠, respectively, as depicted in Fig. 10. Let us consider the application of a single moment component at a time. When $Mx$ is applied to the end-effector, limbs 2 and 3 resist the rotation, but limb 1 does not resist the rotation since all the R joints in the limb are passive. Similar behavior applies when $My$ and $Mz$ are separately applied to the end-effector. Table 5 summarizes this behavior. Accordingly, a limb only resists two moment components as shown in Fig. 12 as it cannot resist the moment component working about an axis parallel with the R joints' axes. Table 6 summarizes the resisted moments in each limb. Comparing the behavior of the manipulator against forces with that against moments, it can be concluded that a force component is only resisted by a single limb that moves in a plane perpendicular to the force, whereas a moment component is resisted by two limbs whose R joints' axes are not parallel with the moment axis.

Based on the behavior of the manipulator's limbs under a force and a moment applied to the manipulator's end-effector as discussed in Secs. 5.1 and 5.2, the stiffness of the entire manipulator can be evaluated through multiple separate analyses at the individual limb level. The evaluation is performed in the following steps:

Step 1: Given a general force and moment applied to the manipulator's end-effector, decompose the applied force and moment into their Cartesian components, namely $Fx$⁠, $Fy$⁠, $Fz$⁠, $Mx$⁠, $My$⁠, and $Mz$⁠.

Step 2: Solve the kinematics of the manipulator. This is required to obtain the posture of each limb.

Step 3: Analyze the deformation of the resisting limb(s) using any feasible method. Please note that the number of separate analyses depends on the number of force and moment components applied to the end-effector. For example, only one analysis is required if only one force component or one moment component exists. At maximum, nine analyses are required in the presence of three force components (⁠$Fx$⁠, $Fy$⁠, and $Fz$⁠) and three moment components (⁠$Mx$⁠, $My$⁠, and $Mz$⁠).

As shown in Table 3, the limb tip out-of-plane deformation evaluation due to forces only considers one force component. Similarly, only one moment component should be used in the evaluation of each limb, although Table 6 shows that two moment components are resisted in each limb. This is to avoid using a moment component more than once, because the deformation of the manipulator's end-effector is obtained by summing the end-effector's deformation obtained from all the limbs. Observing limb 1 in Fig. 12, we notice that $Mz$ bends the limb whereas $My$ rotates the whole limb about the Y-axis. It was observed from the simulation that the end-effector deformation due to the bending moment $(Mz)$ is more dominant. Hence, it is used in the evaluation of limb 1. Similarly, $Mx$ and $My$ are used in the evaluation of limbs 2 and 3, respectively.

Step 4: Sum the deformations from all the analyses to obtain the deformation of the end-effector due to all the force and moment components working at the end-effector.

The four aforementioned steps only require the assembly of the links in each limb. The assembly of all the limbs into a complete manipulator is not required. Although this approach requires multiple individual analyses, each analysis is simpler, more intuitive, and easier to verify since it is performed in the individual limb that involves a small number of links. This approach also provides better knowledge of the behavior of the manipulator, such as its end-effector deformation, under each load component.

Consider a 3PRRR parallel manipulator with proximal and distal links in all the limbs that have identical lengths of 250 mm. All the links have a square cross section of 30 × 30 mm and are made of aluminum with the Young modulus E of 71.1 GPa and the shear modulus G of 26.7 GPa. The end-effector offset in the X, Y, and Z directions is equally 80 mm. With the manipulator at a pose of $x=y=z=350mm$⁠, forces $Fx=Fy=Fz=−50N$ and moments $Mx=My=Mz=25Nm$ are separately and altogether applied to its end-effector.

Consider force components $Fx=Fy=Fz=−50N$ are applied to the end-effector. Table 7 shows the end-effector deformation due to separate force components, evaluated at the limb level. The last line of the table shows the end-effector deformation if all the force components are applied to the end-effector. It is obtained by simply summing the end-effector deformations due to the separate force components, without a need to assemble the complete manipulator.

Table 8 shows the deformation of the end-effector computed using various methods, including the simplified approach proposed in this article. The table shows that the deformation computed using the proposed approach is close to that computed using the assembled manipulator model.

Figure 13 shows the deformation of the 3PRRR manipulator in the MSA model. Figure 14 shows the meshed manipulator in the ansys fea model.

Figure 15 shows the deformation of the manipulator's end-effector across its workspace, evaluated in the MSA model. Figure 16 shows a possible useful workspace with minimal deformation that represents the optimal stiffness. The stiffness map shows that the optimal stiffness is achieved at positions far enough from the workspace boundaries. In other words, the stiffness is poor at positions around the workspace boundaries. This can be related to the singularities. Since the singularities of the manipulator occur along the workspace boundaries, the stiffness of the manipulator is poor at singular positions or close to the singular positions.

Now consider moments $Mx=My=Mz=25Nm$ are applied to the manipulator's end-effector. The moment components $Mz$⁠, $Mx$⁠, and $My$ are respectively applied to limbs 1, 2, and 3 in the limb evaluation. The end-effector of the complete manipulator due to all the moment components can be approximated by summing the deformations obtained from the limbs' evaluation, as shown in Table 9. A comparison between the solution using this approach and that using an assembled manipulator model is shown in Table 10.

Now consider forces $Fx=Fy=Fz=−50N$ and moments $Mx=My=Mz=25Nm$ are applied to the manipulator's end-effector. The evaluation of each limb can be performed by combining the evaluations against forces and moments presented earlier. The end-effector of the complete manipulator due to all the force and moment components can be approximated by summing the deformations obtained from the limbs' evaluation, as shown in Table 11. A comparison between the solution using this approach and that using an assembled manipulator model is shown in Table 12.

To investigate the influence of the manipulator's kinematic parameters on the manipulator's stiffness, a global stiffness index will be used to measure the stiffness quality at various parameter values. The global stiffness index is the mean of the average end-effector deformations across a specified workspace. This index will be evaluated based on the MSA model of the manipulator.

Figure 18 shows the end-effector deformation changes with the link length ratio $L1/L2$⁠. The plotted curves indicate the means of averaged end-effector deformations over the changing ratios at some fixed values of $L1$ or $L2$ at a time. The dashed curves correspond to the cases in which $L1$ is fixed at some values while $L2$ varies from 150 mm to 300 mm. On the other hand, the solid curves correspond to the cases in which $L2$ is fixed at some values while $L1$ is varied from 150 mm to 300 mm. The plot shows that the value of the fixed link lengths affects the best ratio, i.e., the ratio resulting in the minimum end-effector deformation. An observation of the whole curves shows that a ratio of 1 is in general a tradeoff.

In practice, the end-effector cannot be zero. Figure 19 shows that shorter end-effector offset distance leads to less mean averaged end-effector deformation. The deformation shown in the figure is based on modeling the end-effector as three orthogonal beams connected by fixed joints at the end-effector position (x, y, z). All of the beams have an identical length that represents the end-effector offset distance.

An investigation on the stiffness of a two-link limb subject to out-of-plane loads was conducted. We proposed an analytical solution of a two-link limb subject to an out-of-plane moment to complement a previously published analytical solution of a two-link limb subject to an out-of-plane force. We verified that the analytical solution is identical to the MSA solution, while both solutions have a slight difference from the FEA solution. Furthermore, this study investigated the effect of the limb angle and the links' lengths on the limb's out-of-plane stiffness. The following can be concluded from the parametric analysis of the two-link limb subject to an out-of-plane loading:

• The closer the limb is to the folded configuration, the stiffer the limb in the out-of-plane direction. This corresponds to a shorter effective length of the limb.

• The effect of the link length can be evaluated based on the resulting effective length. The shorter the effective length, the stiffer the limb, and vice versa.

• A larger link length ratio $L1/L2$ with constant total length and effective length results in a higher out-of-plane stiffness.

The stiffness evaluation of the two-link limb serves as a base for the subsequent stiffness evaluation of a decoupled Cartesian parallel manipulator. A new approach to quickly estimate the stiffness of the entire manipulator without the need to assemble the limbs of the manipulator shows results close to those obtained by using the common approach that requires the assembly of the whole manipulator. The new approach only evaluates the deformation of each limb independently, followed by superposing the deformations of all the limbs. The new approach can be used utilizing any existing deformation-based stiffness evaluation method such as analytical method, MSA, and FEA. Not only the new approach provides a quick stiffness evaluation as it avoids an assembly process, but it also provides a direct insight into the contribution of the limb stiffness to the entire manipulator stiffness. Furthermore, a parametric analysis was performed on the decoupled parallel manipulator. The parametric analysis shows that shorter links' lengths and end-effector offset result in a stiffer manipulator while a link length ratio of 1 is the best tradeoff for the manipulator stiffness.

• This work was supported by Khalifa University of Science and Technology under Grant No. RIG-2023-003.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.