## 1 Introduction  
Compliance is the term used when referring to low levels of stiffness, in other words, when flexibility is the characteristic feature. Compliant systems can be categorized in compliant mechanisms (Figure 1a), compliant joints (Figure 1b), and flexure elements (Figure 1c). Compliant mechanisms transmit motion, force, and energy through the elastic deformation of flexible members known as flexures. Compliant joints are the key-components in compliant mechanisms; they act analogously to the kinematic pairs in rigid-body mechanisms. The basic form of a compliant joint is a single flexure, which is usually formed by removing material from a bulk piece. A compliant joint can be composed of more than one flexure and could be considered as a compliant mechanism on its own. Flexures stretch, bend, or twist during deformation, which leads to the relative motion between parts in a compliant system. Figure 1: Compliant systems: (**a**) compliant mechanism, (**b**) compliant joint, and (**c**) flexure. Compared with traditional rigid-body mechanisms, compliant mechanisms have many inherent advantages, including elimination of sliding contact between surfaces, monolithic manufacturing, higher precision, reduced friction, and reduced weight [1]. In contrast to their rigid-body counterparts, compliant mechanisms facilitate smooth, continuous motion, without backlash, and with high levels of repeatability and precision [2]. Therefore, compliant mechanisms find applications in fields requiring high precision motion, such as precision manufacturing [3], biological cell handling [4], micro/nano-manipulation [5], positioning stages [6], microelectromechanical systems (MEMS) [7], micro-vibration suppression [8], optical guides [9], and so forth. Apart from the advantages, compliant systems have some drawbacks owing to their intrinsic coupling of kinematics and elastomechanical behaviors. Consequently, design and analysis cannot be done by separating kinematics and dynamics. The design process is even more complicated if the compliant elements undergo large deflections, due to the resulting nonlinear relationship between stress and strain. The challenging process for designing and analyzing compliant systems limits the wide use of three-dimensional (3D) architectures with multi Degrees of Freedom (DoFs),restricts the design to small deflections, and may lead to inaccuracies such as cross-coupling errors and erroneous undesired displacements termed as parasitic motions [10; 11; 12; 13]. Applications increasingly demand compliant systems with large workspaces, multiple-DoFs, low cross-coupling errors, and reduced parasitic motions. Attempts to reduce inaccuracies and limitations have been proposed, some examples are the use of redundant constraints [14; 15], compound joints [16; 17], addition of subsystems [18; 19], implementation of tracking control systems [20], and design of spatial complex architectures to perform basic planar motions [21]. However, all of these techniques are based on the incorporation of elements or sub-systems to the compliant system, thus increasing complexity, size, and weight. The performance of a flexure element, i.e., its range of motion, accuracy, kinematics, and dynamics, is determined by the proper characterization of its force-deformation relation, i.e., the stiffness (or compliance) of the element. Having control over the stiffness of the flexure elements appears as an ideal way to deal with inaccuracies and limitations of compliant systems. As the stiffness of a flexure element is governed by the material that composes it, boundary conditions and external geometry, changing its dimensions will result in a modification to its stiffness. However, the geometry is generally dependent on the available material stock, manufacturing process, or other design constraints [22]. Moreover, a change in the overall dimensions of a flexure element may lead to a loss in the off-axis stiffness leading to instability, thus resulting in inaccuracies.