## 3 Tailoring Stiffness via Topology Optimization
#### Topology Optimization in Compliant Systems as Flexure Elements  
With the potential of Topology Optimization to design compliance materials already being stated, the works using this approach in compliance systems at flexure level are reviewed. For example, Topology Optimization has been used to design optimal compliant kinematic limbs that are part of larger compliant mechanisms [200; 201; 202]. A schematic  
Figure 11: Process to tailor stiffness using Topology Optimization applied to cellular design domains that leads to functionally graded Cellular Materials. Figure 10: Process to tailor stiffness using the multiscale process that combines Topology Optimization with Cellular Materials. description of this process applied to the limbs of 2D flexure-based compliant parallel mechanisms is shown in Figure 12. [PERSON] et al. [200] performed a Topology Optimization process to synthesize and optimize the topology, shape, and size of the limbs in a flexure-based compliant parallel mechanism (see Figure 12a), resulting in a significant improvement in the mechanism's workspace dimensions and increments in the mechanism's stiffness ratios, i.e., ratios of off-axis to actuating stiffness. Another work that dealt with the optimization of compliant kinematic limbs was that of [PERSON] et al. [201], where those of 2D compliant parallel mechanisms were regarded as separate design domains for a Topology Optimization problems (see Figure 12(b-i,c-i), and the Jacobian matrix was introduced into the field of Topology Optimization. This resulted in compliant parallel mechanisms with maximized workspaces that have compliant kinematic limbs accurately characterized facilitating the mechanism analysis. Later, [PERSON] et al. [202] enhanced their previous research by including an improved stiffness evaluation generating new optimal compliant limbs and thus compliant parallel mechanisms (see Figure 12(b-ii,c-ii) that have better levels of off-axis stiffness, and are similar to the generated in their previous work. Most recently, a different technique, not applied at the flexure level, has been used to design the limbs of compliant parallel platforms; this is based on keeping the mobile and fixed platforms as rigid bodies while applying a Topology Optimization process on a design domain that connects both platforms, examples using this technique designing 2D and 3D compliant parallel platforms can be found in [203, 204, 205, 206], respectively. Topology Optimization of single flexures, i.e., those that are not part of a compliant mechanism, has been also addressed. The most commonly studied single flexures are the Single-Axis Rotational Flexure Joints (SARF)). Studies of Topology Optimization used in the design of SARF) can be categorized according to the geometry of the design  
Figure 12: Schematic description of the Topology Optimization process to design the optimal compliant limbs of 2D compliant parallel mechanisms: (**a**) 3-DoF (\(x,y,\theta\)) compliant parallel mechanism proposed by [PERSON] et al. [200] (Adapted from [200] 2015, Elsevier); subfigures (**b,c**) are the 2-DoF (\(x,y\)) and the 3-DoF (\(x,y,\theta\)) compliant parallel mechanisms, respectively, proposed by [PERSON] et al. found in (i) [201] (Adapted from [201] 2016, Elsevier); and (ii) [202] (Adapted from [202] 2018, ASME). domain used in the process. Initial attempts were performed using 2D rectangular design domains as shown in Figure 13. Examples using this approach are the works of [PERSON] et al. [207, 208, 209]. In [207], [PERSON] et al. performed the design and optimization of SARFJ, searching to maximize the compliance in the desired direction while minimizing the compliance in the other directions. The design problem was modeled and analyzed as a small-displacement problem using linear Finite Element Analysis (FEA) resulting in the final design shown in Figure 13a.