# Dynamic Modeling of McKibben Muscle Using Empirical Model and Particle Swarm Optimization Method
## 1 Introduction
As suggested by [PERSON] in 2000, [PERSON] in 2009 and [PERSON] in 2011 also considered implementation of correction factor \(q\)\(>\)1 for shape irregularity [17; 18]. The correction factor \(q\) is a pressure dependent parameter of exponential function. Afterwards, [PERSON], in 2012-2015, improved the existing empirical model proposed by [PERSON] and [PERSON] with 5-6 parameters models [19; 20]. On contrary, [PERSON] in 2013 proposed an advanced geometric muscle model where changes of the pressure and diameter of the muscle can be varied.In addition, volume of cylinder was expressed as function of muscle diameter and valve diameter [21]. Similar to this work, [PERSON], in 2014, also considered the same issues where the effects of the end-cap diameters were based on cone shapes [22]. On the other hand, [PERSON] in 2014 evaluated the static muscle force based on empirical stiffness to actuate the muscle actuator via hydrogen absorption/desorption [23]. Inspired by the [PERSON] model, [PERSON] in 2016 proposed a synthetically empirical model consisted of hysteresis, viscous damping, rubber elasticity and contractile elements in parallel [24]. Recently, [PERSON], in 2016-2018, also considered an exponential correction factor \(q\) to account for shape irregularities of McKibben muscle [25; 26; 27]. These researches were focused on quasi-static empirical modeling without quasi-rate and historic dependencies. Eventhough, [PERSON] considered both quasi-static and historic dependencies by creating minor hysteresis loops. Furthermore, [PERSON] contemplated quasi-static, quasi-rate, and historic dependencies in their model. In addition, [PERSON] used correction factor \(\mu(\varepsilon)\) to compensate the asymmetric behavior of McKibben muscle at low and high pressure; while [PERSON] implemented the 2 nd OP equations within the region. Similarly, [PERSON] introduced empirical model with 5-6 parameters to handle this asymmetric region during low and high input pressure. However, most of others researchers just implemented intelligent controllers such as PID, Cascaded, Tracking, Fuzzy Logic, and Fuzzy Adaptive PID to compensate the nonlinearities of McKibben muscle. On the other hand, dynamic modeling of hysteresis considers the output's delay which happened during cyclic contraction and expansion of McKibben muscle. This is due to frictions in between thread-to-thread and thread-to-bladder of the McKibben muscle including other unidentified factors. When compared static and dynamic models of McKibben muscle, asymmetric hysteresis offset can be observed. To resolve this behavior, [PERSON], in 1988, proposed a model consisted of frictional term proportional to the angular velocity of the pulley and spring term proportional to the pulley angle [7]. Then, [PERSON], in 1996, implemented static coulomb friction model and create static offset during both contraction and expansion stages. While, [PERSON], in 2000, proposed and compared the implementation of static and dynamic coulomb friction models [28; 29]. Afterwards, [PERSON], in 2001, resolved the problem by using three elements models of spring, viscous damper and coulomb friction [30]. On contrary, [PERSON], in 2003, implemented static offset of force which is practically similar to the coulomb friction model [31]. On the other hand, [PERSON] in 2006 considered the dynamic modeling through the FLP graph implementation [32]. Following [PERSON], [PERSON], in 2006, introduced dynamic coulomb friction model. However the proposed empirical model neglected the \(k\) factor [12]. Furthermore, [PERSON], in 2009, implemented both velocity's dependence coulomb and viscous friction models [17]. While, [PERSON] in 2010 applied static coulomb friction model as proposed by [PERSON] and [PERSON] [33]. Then, [PERSON] in 2011 introduced a spring element under no pressure condition into coulomb and viscous friction models proposed by [PERSON] based on vertical load system [18].