**1.1. Design methodology review for SMA actuators**

Many SMAs applications have been studied in recent years. Kuribayashi [5] proposes a rotary joint based on a bending SMA actuator, while, in [6] design and applications of SMA

actuators are presented. Microrobots can be developed using SMA as shown in [7] where there is a basic method to design the SMA spring based on a thermo-electromechanical approach. Reynaerts et al. [8] present design considerations concerning the choice of the active element to evaluate SMA actuator efficiency. Lu et al. [9] design a high strain shape memory actuator taking into account pseudoplasticity and compared its performance with traditional actuators. Due to SMA high non linearity, design curves relying on experiments are proposed in [10] to assess the SMA actuator geometry. A comprehensive review of applications of SMAs in the field of mechanical actuation has recently been published in [11]. Jansen et al [12] develop a linear actuator used as a drive module in an angular positioning mini-actuator. This architecture allows both large force and long strokes to be obtained. Strittmatter et al. [13] propose a SMA actuator for the activation of a hydraulic valve, biased by a conventional spring. Bellini et al. [14] propose a linear SMA actuator able to vary the air inflow for internal combustion engines, improving gas combustion and leading to higher efficiency. Haga et al. [15] propos a mini-actuator to be used in Braille displays. Elwaleed et al. [16] develop a SMA beam actuator able to amplify the SMA actuator strain using elastically instable beams. Among the proposed actuator there is lack of simple design instruments to provide basic information to the designer, either due to specific constraints of application or due to the high complexity of the thermomechanical material models used. In order to answer for an analytical design methodology, the author described in several technical publications a set of equations useful for linear [18] and rotary application [19]. The authors developed the design equations both for SMA actuators under a general system of external forces [20] and produced the design formulae to increase the output stroke thanks to negative stiffness compensators [21]. Moreover two peculiar systems were designed and developed: a telescopic actuator [22] and a wire on drum system [23]. The present work reviews and improves the design rules developed by the authors and set them in a coherent formal analytical framework. Design examples are provided to illustrate the step-by-step application of the design optimization procedures for realistic case studies.
