# Dynamic Modeling of McKibben Muscle Using Empirical Model and Particle Swarm Optimization Method
## 1 Introduction  
McKibben muscle plays an important role in the development of assistive rehabilitation robotics system. The development of rehabilitation orthosis, which implemented McKibben muscle, is rather slow when compared to other types of actuated rehabilitation orthosis using ac-motor, dc-motor, pneumatic cylinder, linear actuator, series elastic actuator (SEA), and brushless servomotor. However, due to its clear advantages such as low weight, structural flexibility, compactness, and inherent compliance compared to other types of artificial muscles, the researcher's interest in this field has grown exponentially in past decade [1; 2; 3]. This growth also due to the challenges it provides in resolving nonlinearity behaviors and difficulties in controlling such systems. The inherent nonlinearity behaviors of McKibben muscle present because of nonlinear relationship in between its contracting force, contracting length and pressure [4; 5; 6]. The other factors include nonlinearity behaviors of pressure build-up and hysteresis due to its geometric construction and frictions. Various approaches and methods have been introduced to reduce the effect of hysteresis, where a precise control system is desired. In addition, numerous control strategies were considered to resolve nonlinearity problems inboth static and dynamic modeling. Throughout the years, a number of physic and phenomenological approaches had been considered in static modeling such as force generation equation, empirical model, conservation of energy (virtual work), polynomial-based model, and [PERSON]'s muscle model [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39]. However, empirical modeling was often used to characterize the behaviors of McKibben muscle compared to others due to previous achievement [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. This is also similar for the virtual work theorem which can be related with empirical model [28]. On contrary, dynamic modeling of McKibben muscle also have been evolved rapidly with implementation of various models such as coulomb friction and viscous friction [17; 18]. In static modeling of McKibben muscle using empirical model, it was first introduced by [PERSON] in 1988 [7]. After that, [PERSON], in 1994, imposed similar approach and modified the model with empirical constant \(k\) to account for end deformation of the muscle [8]. These early researches became the pillar of other researches and major contribution in this research field as [PERSON] in 2003 introduced empirical dynamic force model with parallel arrangement of a spring element, damping element, and contractile force element [9]. While, [PERSON], in 2005, improved the empirical model with force modeling and considered a one-way cylinder that moves against a spring [10]. The pulling forces of the muscles were approximated by polynomial functions. Meanwhile, [PERSON], in 2006, improved the empirical model and included braid effects on contractile range [11; 12]. Then, analyses its frictional effect to produce a more accurate model and removed the necessity of correction parameter \(k\). Contrarily, [PERSON], in 2006, proposed a combination of empirical-geometric model with a variable spring and parallel variable damper system [13]. Due to its nonlinearities, the static characterization of McKibben muscle becomes more complicated with introduction of these compensation methods. To reduce the complexity, [PERSON], in 2009-2010, introduced empirical modeling using spring system with variable stiffness of parameter K, where the parameter was taken as 2 nd Order of Polynomials (OP) of pressure and length [14; 15]. Then, [PERSON], in 2010, improved the force modeling of McKibben muscle and considered one-way cylinder which moves against a spring. The pulling forces of the muscles were approximated by polynomial functions [16].