Topology optimization of beam-based flexural pivots with arbitrary centers of rotation

As mentioned in Ref. [30], flexural pivots typically support off-axis forces while allowing rotation about the intended axis. Flexural pivots can be categorized as concentrated flexural pivots and distributed flexural pivots. The flexural pivots with concentrated compliance include notch hinges, while flexural pivots with distributed compliance consist of curve-beams, leaf-springs, tape-springs, and cross-spring pivots, which are all beam-based flexural pivots [30]. While structural optimization methods have been effectively used to obtain optimal configurations for notch hinges [19], they are less suited for beam-based flexural pivots, which may limit their applicability in scenarios that require large displacements, low stiffness, or fixed centers of rotation [31]. In this work, the optimization of beam-based flexural pivots is based on a design domain of ground structures [17]. The single-layer ground structure, made up of beams, is more adaptable for beam-based flexural pivots, as depicted in Fig. 2.

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The flexural pivots allow in-plane motion of the right rigid body relative to the fixed left rigid body, as illustrated in Fig. 2a. There is a target center of rotation \(O_L\) on the left rigid body and \(O_R\) on the right rigid body, which overlaps with the initial configuration. When a prescribed rotation \(\theta\) is applied to the right rigid body, as shown in Fig. 2b, an axis shift \(\Delta\) occurs between \(O_L\) and \(O_R\).

For the single-layer ground structure, beams are used to fill the design domain. The beams are connected through nodes; the nodes at the left end of the beam structure are attached to the left rigid body, which is fixed, while those at the right end are connected to the right rigid body, which is movable. At most, eight beams are connected together through one node. The number of beams in one direction is defined as the grid density n. Then, the total number of beams isAlthough both the Geometrically Exact Beam (GEB) formulation and the Absolute Nodal Coordinate Formulation (ANCF) [32] can capture large beam deformations, we employ the GEB formulation by Simo and Vu-Quoc [33] in this optimization due to its additional capability to model shear deformation.

For an ideal revolute pivot, the axis shift \(\Delta\) of the center of rotation in the deformed configuration should be zero. Simultaneously, it should demonstrate high off-axis stiffness and very low rotational stiffness [11]. However, optimizing based solely on axis shift \(\Delta\) can lead to uncertainty in rotational stiffness and off-axis stiffness. Therefore, both stiffness and axis shift \(\Delta\) should be treated as objectives in the optimization process. Notably, the axis shift \(\Delta\) may lead to extra rotational stiffness around the target center of rotation. To simplify the optimization objectives, we use the rotational stiffness around the target center of rotation as a measure to minimize both the axis shift and the rotational stiffness. To address this, two objective functions are formulated to achieve the desired properties of maximal off-axis stiffness and minimal rotational stiffness around the target center of rotation. The stiffness-based objective function \(\Phi _s\) incorporates a combination of stiffness metrics, considering both rotational and off-axis stiffness. Ideally, the flexural pivots should exhibit low rotational stiffness around the target center of rotation while maintaining high off-axis stiffness. In addition, as mentioned in Ref. [2], the strain energy-based objective function \(\Phi _e\) considers the strain energy of all beams under translational and rotational movements. Both objective functions are elaborated in the following sections.

For flexural pivots, the ideal stiffness should exhibit very high off-axis stiffness and very low rotational stiffness to provide a mechanical advantage, as mentioned in Ref. [2]. To obtain a lightweight structure, traditional optimization of compliant mechanisms uses the target volume or volume fraction as a constraint [2]. For a problem without a target volume, extended optimization is used, as mentioned in Ref. [34]. For this purpose, a stiffness-based objective function \(\Phi _s\) is introduced to optimize both the rotational stiffness and off-axis stiffness, while also considering the total volume, thereby formulating it as a multi-objective optimization problem, which is expressed aswhere w is a weight factor that can balance the ratio of rotational stiffness and off-axis stiffness [11]. When w is close to zero, high off-axis stiffness becomes the primary consideration. Conversely, when w is close to 1, low rotational stiffness becomes more critical. However, this factor must be appropriately chosen; otherwise, it may lead to undesirable results. In addition, the rotational stiffness around the target center of rotation is \(C_r\), and \(V_{T\!ol}\) is the total volume of beams. To compute the stiffness, as illustrated in Fig. 3, the off-axis stiffnesses \(C_x\) and \(C_y\) are considered, where the coupling stiffness between the two orthogonal directions is not included in the off-axis stiffness. The reference frame is consistent with Ref. [12], where the horizontal and vertical directions are denoted as the x- and y-axes, respectively. To minimize the off-axis stiffness, the inverses of \(C_x\) and \(C_y\) are used to drive the objective function toward zero, and their squares are taken to compute the quadratic mean. This approach avoids negative values, which may introduce numerical instability or result in negative infinity during computation. Additionally, \(V_{T\!ol}\) is used as a multiplicative term, following the approach referenced in Ref. [34].

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A configuration with an ideal joint, as depicted in Fig. 3a, is used to calculate the rotational stiffness around the target center of rotation. In this configuration, a revolute joint is positioned at the target center of rotation and prescribed with a specific rotation angle \(\theta\). Under these circumstances, the reaction torque \(T_r\) can be determined and the rotational stiffness \(C_r\) isFig. 3b shows a configuration without an ideal joint, which is used to calculate the off-axis stiffness. To determine the off-axis stiffness, displacements \(D_x\) and \(D_y\) are prescribed on the right rigid body along the x- and y-directions, as shown in Fig. 3b. The corresponding reaction forces \(F_x\) and \(F_y\) are then computed. Thus the off-axis stiffness can be obtained withandFor topology optimization, the SIMP method is utilized [2], which uses the relative density as a variable to adjust the material properties during optimization. The Young’s modulus \(E_i\) of each beam \(i\in ~\{1,2,..., \bar{N}_b\}\) is defined as a variable property, which will be changed by the relative density \(\rho _i \in [0,1]\) of each beam i, aswhere \(E_0\) is the Young’s modulus of solid material. To achieve the range of \(\rho _i\), an approximated Heaviside function is introduced [35], denoted aswhere \(\xi _i\) is a target variable with \(\xi _i \in \left[ -1, 1\right]\) and \(\beta\) is the parameter used to control the slope of the density change, whereas a larger \(\beta\) gives a narrower band of intermediate densities, see Ref. [36]. Therefore, the total number \(\bar{N}_v\) of variables in the single-layer ground structure isThe total volume \(V_{T\!ol}\) is defined aswithwhere \(L_i\) is the length and \(A_i\) is the cross-sectional area of beam i. However, when the relative density \(\rho _i\rightarrow 0\) for all variables, \(\Phi _s\) converges to 0, which is an undesired optimized result. Consequently, a constraint condition of the relative density \(\rho _i\) is defined aswhich implies that the minimum number of beams required to connect two rigid bodies is n. Since the number required for a leaf spring is n, using fewer beams would result in a weak connection between the two rigid bodies, which is problematic when minimizing the total volume [34]. To normalize the stiffness-based objective function \(\Phi _s\), the stiffnesses \(C_{x0}\), \(C_{y0}\), and \(C_{r0}\) of the ground structure with initial variables \(\xi _{i0}=0\) are employed as a reference stiffnesses. Therefore, Eq. (2) can be rewritten aswhere the only control parameter is the weight factor w.

In the optimization of compliant mechanisms, the objective function typically uses strain energy as the target, while the volume is set as a constraint [2]. For the optimization of flexural pivots, both the strain energy of translational movements in various directions [19] and the strain energy of rotational movement are considered [20]. The objective function is defined to minimize the strain energy within a target volume. On the basis of this concept, both translational and rotational strain energy are considered in this paper. Additionally, the volume is introduced as an optimization target, as shown in Eq. (2). Consequently, the strain energy-based objective function \(\Phi _e\) is described aswhere \(U_r\) is the strain energy of all beams under a certain rotation angle, while the right rigid body and left rigid body are connected with an ideal revolute joint at the target center of rotation, as depicted in Fig. 3a. \(U_x\) and \(U_y\) represent the strain energies of all beams when a specific displacement is prescribed at the target center of rotation. During these computations, there is no direct connection between the two rigid bodies, see Fig. 3b. Generally, \(U_*\) is used to denote the strain energy under different deformation conditions. Since GEB elements are used, the strain energy \(U_*\) can be computed aswhere, \(\varepsilon _i\) is the tensile strain, \(\kappa _i\) is the curvature and \(\gamma _i\) is the shear strain of beam i. The shear modulus \(G_i\) iswith Young’s modulus \(E_i\) using Poisson’s ratio \(\nu\) of each beam i. In addition, \(L_i\) is the length of beam i, A is the area of the beam’s cross-section, and I is the second area moment, which has uniform geometric properties. With the design domain and proposed objective functions, the detailed optimization model and approach are explained in the next section.

Compared with the stiffness-based objective functions in Eqs. 2 and 12, Eq. 13 does not include any weight factors or normalization. To serve as a comparative bridge between two different objective function concepts, an reference stiffness-based objective function \(\bar{\Phi }_s\) is introduced, asThis function is intended to directly highlight the distinction between the stiffness-based and strain energy-based approaches.

The single-layer ground structure, described in Sect. 2.1 and depicted in Fig. 2, employs one layer of ground structure as the design domain, restricting the results to a single flexure joint. More generally, to achieve multileaf flexure joints [30], such as cross-spring pivots, we propose a dual-layer ground structure for beam-based flexural pivots, as illustrated in Fig. 4. This approach enables the design of multileaf flexure joints with various topologies. The connectors \(c_j\) at corresponding positions in the two layers are utilized as fixed joints, which are also defined as variables.

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The grid density n, is defined as the number of beams along the length and width, and it directly affects the number of target variables \(\xi _i\). Therefore, the total number \(\bar{\bar{N}}_v\) of variables in the dual-layer ground structure iswhich contains the total number \(\bar{\bar{N}}_b\) of beamsand the total number \(\bar{\bar{N}}_c\) of connectorsAs shown in Fig. 4, if the target center of rotation is on the x-axis, the layer-1 and layer-2 variables are symmetrical with respect to the x-axis, meaning a single set of variables can describe both layers’ structures. Thus, to increase computational efficiency, the total number \(\bar{\bar{N}}_v^*\) of variables of a symmetrical dual-layer ground structure can be reduced toand the corresponding number \(\bar{\bar{N}}_b^*\) of beams is given byNote that the two layers are modeled in a single plane in Fig. 4, although they are illustrated with a gap between them for clarity.

On the basis of the flexural hinge investigated in Ref. [1], the dimensions and material properties are given in Table 1.

The connectors between the two layers are parameterized with a target variable \(\eta _j \in \left[ -1,1\right]\), where \(j \in \{1,2,...,\bar{\bar{N}}_c\}\). The connector \(c_j\) is valid and marked as "True", when the target variable \(\eta _j\) is positive; otherwise, the connector \(c_j\) is not valid and therefore marked as "False".

Overall, for both the single-layer and dual-layer ground structures, the optimization problem can be expressed as:where, both the stiffness-based objective function \(\hat{\Phi }_s\) from Eq. (12) and the strain energy-based objective function \(\Phi _e\) from Eq. (13) are employed for the optimization. The target variable \(\xi _i\) contains \(\bar{N}_b\) variables for the single-layer ground structure, \(\bar{\bar{N}}_b\) variables for the general dual-layer ground structure, or \(\bar{\bar{N}}_b^*\) variables for the symmetrical dual-layer ground structure. Additionally, the target variable \(\eta _j\) contains \(\bar{\bar{N}}_c\) variables, which are used only for the dual-layer ground structure.

For multi-objective optimization problems, nondeterministic methods such as genetic algorithms are widely used because of their effectiveness [28, 37]. This approach progressively generates increasingly refined designs. We use the approach proposed in Ref. [37], which uses a range reduction factor to demonstrate higher accuracy with lower computational cost. In the genetic algorithm, the initial variables are randomly generated within a specified range. After each generation, the surviving populations undergo crossover and mutation as illustrated in Fig. 5. In each generation, individuals with smaller objective function values are more likely to be selected for reproduction and crossover, resulting in a population with progressively better designs.

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The genetic algorithm parameters set in Fig. 5 include the mutation reduction factor, crossover probability and distance factor, which are defined in Ref. [37]. The population size and number of generations are related to the number of beams \(\bar{N}_b\) or \(\bar{\bar{N}}_b\), which impacts both computational efficiency and accuracy, as discussed in the subsequent examples.

The process of obtaining the stiffness and strain energy involves three steps in the static model, see Fig. 5. In Step 1 and Step 2, the off-axis stiffnesses are calculated for prescribed displacements of \(D_x\) = 1 \(\mu\)m and \(D_y\) = 1 \(\mu\)m, as shown in Fig. 3b. Thus, the off-axis stiffnesses \(C_x\), \(C_y\), and the strain energies of translational movements \(U_x\), \(U_y\), are obtained. Then, in Step 3, considering the rotational stiffness \(C_r\) and the strain energy of rotational movement \(U_r\), a revolute joint is added at the target center of rotation, see Fig. 3a, with a prescribed rotation angle \(\theta\) of \(\hbox {10}^\circ\). At this angle, the stiffness exhibits nonlinear behavior, as demonstrated in Ref. [38], and can therefore be considered a basis for identifying large deflection. For the ground structure model, each beam is discretized into 10 GEB elements, as shown in Fig. 6a. For the approximated Heaviside function in Eq. (7), we set \(\beta = 16\), as described in Ref. [35]. Thus, by employing the objective functions, the fitness of all individuals in the population is evaluated. Before the optimization, the parameters in Eq. 12, \(C_{x0}\), \(C_{y0}\), and \(C_{r0}\), are computed with all \(\xi _i=0\) and \(\eta _i=0\).

Note that random parameters can easily lead to numerical failures in large deflection, as also noted in Ref. [39]. The models are established within the framework of the multibody dynamics simulation code Exudyn¹ [40]. Each static deformation analysis presented in this paper is divided into 10 load steps.

A benchmark model is established with the target center of rotation at the center of the design domain, as shown in Fig. 6. This model allows the investigation of how two objective functions and their parameters influence the results. The model is established as a single-layer with a grid density of \(n=2\), as illustrated in Fig. 6a. The population size is set to \(100\cdot \bar{N}_b\) and optimized for 30 generations.

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As shown in Fig. 6e, deformed beams clearly have large curvatures, which can lead to nonlinear problems [26]. The transparency of each beam corresponds to its relative density \(\rho _i\). A dark red color indicates a relative density \(\rho _i\) close to 1, a lighter color represents a flexible beam with lower density, and the disappearance of a beam indicates a relative density \(\rho _i\le 0.05\). As shown in Fig. 6b–d, the variation of the weight factor w in Eq. (2) influences the topological configuration of the flexural pivot. As shown in Fig. 6b, a weight factor \(w=\) 0.1 means that the rotational stiffness has a smaller proportion in the stiffness-based objective function \(\hat{\Phi }_s\), while \(w=0.9\) in Fig. 6d gives a clear topology and \(w=0.995\) in Fig. 6c results in a very flexible structure. When w is set to 1, the optimization yields a structure lacking effective connections between rigid bodies, as the rotational stiffness may reduce to zero. When the stiffness \(C_y\) is neglected in Fig. 6f, the resulting configuration evolves into a leaf spring [30]. It can be observed that optimizing the objective function for suitable stiffness produced results resembling a notch hinge [30], as shown in Fig. 6g. This configuration, representing a single ground structure with a grid density of \(n=2\), is denoted as S2. Generally, incorporating a weight factor for direct control of design variables allows the stiffness-based objective function \(\hat{\Phi }_s\) to become a more appropriate solution.

Similarly, the investigation is also applied to a grid density of n = 3 for the ground structure. The corresponding model is shown in Fig. 7a. The population size is set to \(200\cdot \bar{N}_b\) and optimized for 40 generations.

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The same to Fig. 6b, when w = 0.1, the result in Fig. 7b is relatively stiff, exhibiting both high off-axis stiffness and high rotational stiffness. As shown in Fig. 7c, for \(w=0.9\), the result of an optimized stiffness is a cross-spring pivot with two reinforcement beams, which drives the configuration to a high stiffness in \(C_x\) and \(C_y\). Simultaneously, as illustrated in Fig. 7d, \(w=0.995\) yields a typical cross-spring pivot, as the two cross beams remain unconnected. The resulting single ground structure with a grid density of \(n = 3\), is denoted as S3.

The results of strain energy-based objective function in Eq. 13 and the results of reference stiffness-base objective function Eq. 16 is illustrated in Fig. 8. At a grid density of n = 2, optimizing with either the strain energy-based objective function \(\Phi _e\) or the reference stiffness-based objective function \(\bar{\Phi }_s\) consistently produces a leaf spring with extra flexures, see Fig. 8a, b. Likewise, at a grid density of n = 3, the results in Fig. 8c, d show that the beams in the center domain exhibit high flexibility. As none of these analogous results yield an ideal flexure pivot, this underscores the critical role of the weight factor w and the normalization of physical quantities in the optimization process.

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Considering the computational cost of different objective functions shown in Table 2, the time required to evaluate each objective function was measured on a system equipped with an Intel Core i7-11850 H 2.50 GHz CPU.

It can be conclude that the strain energy-based approach incurs a higher computational cost compared to the stiffness-based approach. This is because the latter focuses solely on the force or torque at the output, eliminating the need for detailed stress or strain analysis of individual elements. Therefore, for optimizing beam-based flexural pivots, a stiffness-based objective function offers superior efficiency and can achieve ideal configurations when an appropriate weight factor is applied. Increasing the grid density notably amplifies this time cost, often more than doubling it. The combined impact of grid density, population size, and generations significantly affects the overall time cost. Consequently, the following research will utilize a grid density of n = 2 or n = 3 for \(\hat{\Phi }_s\) in the optimization process.

In the previous section, Figs. 6 and 7 illustrate the optimized design obtained with the stiffness-based objective function \(\hat{\Phi }_s\), described in Sect. 2.2. These designs can be either notch hinges or cross-spring pivots, depending on the presence of a connected node between the beams. In a dual-layer ground structure, defining the connector as a variable can lead to more optimized results. Thus, for the purpose of confirming the dual-layer ground structure, the model with a grid density of n = 2 and n = 3 is investigated. The target center of rotation is set to the center of the design domain, with red beams representing layer-1 and green beams representing layer-2, as illustrated in Fig. 9. The variables of the layers have symmetry in both models, so with the grid density \(n=2\), the population size is set to \(100\cdot \bar{\bar{N}}_b\), and we are using 30 generations. For variables with grid density \(n=3\), the population size is set to \(200\cdot \bar{\bar{N}}_b\) and 40 generations.

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In Fig. 9a, to clarify, the green layer appears slightly wider, although, in reality, all beams have the same thickness t, see Table 1. With reference to Fig. 7d, \(w=0.995\) leads to the results for the cross-spring pivot. As shown in Ref. [4], the optimized result of grid density \(n = 2\) represents an ideal cross-spring pivot, as it lacks connectors between the two layers, thereby validating the effectiveness of the proposed approach. The resulting dual-layer ground structure with a grid density of \(n=2\) is denoted as D2.

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Based on the same weight factor \(w=0.995\), the optimized result of grid density n = 3 includes two additional reinforcement beams combined with a typical cross-spring pivot, as shown in Fig. 10b, which is denoted as D3. In addition, there are two reinforced flexures in both layers with four connectors to tie the nodes together, as shown in Fig. 10c, d. Nevertheless, the topology is the same as the topology shown in Fig. 7c. Considering that the weight factor w represents the ratio between the off-axis stiffness and the rotational stiffness terms in Eq. 16. An ideal double-layered cross-spring pivot configuration can be achieved with a w value of 0.995, which targets an off-axis stiffness approximately 199 times greater to the rotational stiffness term. Notice that, in the optimization, the unit of rotational stiffness is Nm/rad, while the unit of off-axis stiffness is N/\(\mu\)m.

To ensure an accurate comparison of the achieved optimal configurations, reference models were established in ABAQUS. As an example, in Fig. 11a, the rigid body is modeled with solid elements C3D8R, while the flexures are modeled with shell elements S4R. The color of different parts is determined by the material, denoted as \(E_i\) in Eq. (6). The material of the rigid body is set as steel, with an elastic modulus of \(2.1 \times 10^{11}\) Pa. The properties of the flexure beams are assigned according to the optimized results. The left rigid body is fixed, and the right rigid body has constraints on the translational degree of freedom in the z-direction and rotational degrees of freedom around the x- and y-axes. The stiffness values and axis shifts are given in Table 3.

As shown in Table 3, the stiffness and axis shift obtained from the Exudyn simulation align well with the ABAQUS results. The stiffness values from both simulations exhibit an error of less than 5%, confirming the accuracy of the Exudyn model. The slightly higher rotational stiffness in ABAQUS can be attributed to deformations caused by the anti-elastic curving of the shell. For the comparison of different optimal results, the rotational stiffness of the cross-spring pivot (S3, D2, and D3) is lower than that of the notch hinge (S2), while the off-axis stiffness components, \(C_x\) and \(C_y\), exhibit a more balanced distribution. A comparison between single-layer and dual-layer ground structures with a grid density of \(n=2\) shows that the dual-layer configuration performs better. However, the configuration with grid density \(n=3\) introduces additional topological features that enhance both off-axis stiffness and rotational stiffness. The deformed configurations are compared in Fig. 11b–e.

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Both the Exudyn and ABAQUS models clearly show deformation in the beams under rotation. This study thus confirms that applying the SIMP method within a dual-layer ground structure, coupled with a genetic algorithm, can achieve optimized results. Additionally, for this specific problem, a grid density of \(n=2\) can yield optimized results while maintaining a more concise topology.

Based upon the previously mentioned optimization approach, this work aims to confirm that the optimization approach remains effective when the target center of rotation is arbitrary. For this purpose, three examples are established and optimized, which are shown in Fig. 12. First, in example I, the target center of rotation is shifted away from the centroid of the design domain and positioned at a cross point of beams within the ground structure. Then, example II keeps the same target center of rotation position and sets an initial angle to \(\hbox {30}^\circ\) of the right body, see Fig. 12b. In example III, both rigid bodies have an initial angle and the target center of rotation is outside of the design domain. In these examples, we use the stiffness-based objective function \(\hat{\Phi }_s\) for the optimization. The initial configurations are shown in Fig. 12, and the initially referenced stiffnesses can be obtained.

The population size is set to \(200\cdot \bar{\bar{N}}_b\), with 40 generations, as the ground structure is non-symmetric. In addition, the weight factor is set to \(w=0.995\). Therefore, the optimized configurations and resulting topologies are shown in Fig. 13.

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As shown in Fig. 13a, the center of rotation of example I is at the cross point of two beams. The resulting optimized configuration is a cross-spring pivot with some beams acting as a reinforced structure. Similarly, in Fig. 13b, example II shows a cross-spring pivot with two triangle supports since the target center is not local at any cross points. There are no connectors between the two layers. When the center of rotation is placed outside of the design domain, the topologies exhibit significant differences, see Fig. 13c.

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Additionally, Fig. 13 illustrates the obtained configurations with rotations of \(10^{\circ }\) and \(-10^{\circ }\) applied about the z-axis. The figure provides a more detailed view of the deformations, illustrating rotations of \(\pm 10^{\circ }\) about Rz. The deformed configurations show that the centers of rotation are all near the target center. In Example I, the rotational axis shift is minimal, whereas in examples II and III, the axis shift is noticeably larger compared to the values presented in Table 3. Therefore, this study demonstrates that applying the SIMP method to a dual-layer ground structure, in combination with a genetic algorithm, can effectively generate novel topologies.

To clarify, for each optimized configuration, the off-axis stiffnesses, rotational stiffness, and the axis shift \(\Delta\) under a rotation of the prescribed \(\hbox {10}^\circ\) of the right rigid body are given in Table 4.

As shown in Table 4, all three examples exhibit the same order of rotational stiffness. Example I achieves the minimal axis shift and the highest off-axis stiffness. In comparison, placing the target center of rotation outside the design domain leads to an increase in the rotational axis shift \(\Delta\), highlighting the challenge of achieving centers of rotation outside the assembly.