# Dynamic Modeling of McKibben Muscle Using Empirical Model and Particle Swarm Optimization Method
## 4 Parametric Identification of the McKibben Muscle Model
### Particle Swarm Optimization  
Recently, research area that has been receiving an increase attention by the research community, as well as the industry is the optimization algorithm [40]. It is a procedure or numerical method used to find the maximum or the minimum of a function operating with certain constraints [41]. One of the most reliable optimization methods is particle swarm optimization (PSO). It is a computational algorithm technique based on swarm intelligence. This method is inspired based on the observation of social interaction and animal behaviors (i.e., fish schooling and bird flocking). It imitates the way they search for food by emphasizing cooperation and the competition among the entire group population [42]. A swarm consists of individuals, which is called particles and it is initialized with a population of random solutions; where each of these particles represents a different possible set of the unknown parameters to be optimized. The particles will fly around in a multidimensional search space, and then adjust their position according to its own experience and the experience of its neighboring particle [43]. It is designed to efficiently search the solution space by swarming the particles towards  
Figure 2: Hysteresis data at all dependencies: (**a**) McKibben muscle of 300 mm and (**b**) McKibben muscle of 530 mm. the best fitting solution encountered in previous iterations. This, eventually, will be approaching better solutions through the course of the process; then finally converging on a single minimum or maximum solution [44]. The performance of each particle is determined based on a predefined fitness function of the problem. Actually, the implementation of PSO has been reported in many research works in this field. Besides, PSO has been regarded as a promising optimization algorithm due to its simplicity, low computational cost, and good performance [45; 46]. To describe the PSO algorithm, it is initialized with a population of random solutions. These random solutions are called particles, and each of the potential solutions is also assigned with a randomized velocity function [47]. The PSO relies on the information exchange in between particles of the population called swarm. The particles will adjust their trajectories towards their current best solution (fitness), \(P_{best}\). In addition, each particle modifies its trajectory towards the previous best position, \(G_{best}\), attained by any member of its neighborhood as shown in Figure 3. Furthermore, each particle also moves in the search space with an adaptive velocity function. The measurement of fitness evaluates the performance of each particle to determine if the best fitting solution is achieved. During the process, the fitness of the best individual will improve over time and eventually tends to stagnate towards the end of the process; where the stagnation of the process coincides with the successful discovery of the global optimum. Several parameters such as velocity constant, population size, and number of iterations need to be defined before the process. These parameters will decide to a great extent the ability of global minimization. Where, the maximum velocity affects the ability of escaping from local optimization and refining global optimization. Then, the size of swarm stabilizes the requirement of computational cost and global optimization. Table 1 shows the initialized values for the selected parameters. The optimization parameter is realized either when the maximum number of iterations increased or with an attainment of satisfactory fitness value. The fitness value is a reciprocal of the magnitude of the objective function where minimization of objective function was considered. In this paper, the termination criteria considered with the attainment of satisfactory fitness value, which occurred with the maximum number of 10 generations. For each generation, the best among the 50 particles is considered as a potential solution.