# A Multi-Objective Modified PSO for Inverse Kinematics of a 5-DOF Robotic Arm
## 2 Modified PSO and Multi-Objective Modified PSO  
### The Modified Behaviour PSO, m-PSO  
The inertia weight particle swarm optimization, PSO, as in [20], is the most used PSO version. PSO is a swarm optimization technique where a swarm of (k) individuals, particles, are moving in search space looking for an optimal solution of a problem. Any particle, \(x_{i}^{k}\), represents a possible solution of the problem. The local best, \(x_{i}^{k}\), and global best, \(x_{g}\), are evaluated using fitness function, then particle position updates are performed according to Equations (1) and (2). \[v_{i+1}^{k}=w\;v_{i}^{k}+c_{1}\;rn\;(x_{i}^{k}-x_{i}^{k})+c_{2}\;rn(x_{g}-x_{i }^{k}) \tag{1}\]  
\[x_{i+1}^{k}=x_{i}^{+}+v_{i+1}^{k} \tag{2}\]  
In PSO, particle positions are updated toward the best global and the best local solutions with random moderated steps. The particles are evaluated iteratively according to a fitness function. The processing may be stopped iteratively or when a solution satisfies a given precision. PSO will always return a solution, the best one obtained at the end of its processing. Several improvements have being proposed since the early PSO variant, essentially to avoid the premature convergence since PSO uses confused search/exploit phases. In [21], the authors proposed to manage the inertia weight according to population diversity. Several adaptive PSO variants were reviewed in [22; 23]. Adaptive proposals included inertia weight adaptations, as well as the control of the cognitive and social factors, \(c_{1}\), and \(c_{2}\), to ensure convergence. The adaptive capacities of the reviewed method were limited and the majority of survived adaptive versions are as efficient as the inertia weight PSO [23]. Fitness-adaptive PSO are adaptive PSO variants where PSO parameter changes were arranged according to the fitness of the obtained solutions. In [24], the authors proposed to adapt the inertia weight according to the fitness of a given solution and the mean fitness obtained so-far; a maximum inertia weight is applied to particles for which fitness is higher than the mean obtained so far, while a decreasing fitness is used if not. In [25], the authors also proposed to adapt the inertia weight according to the fitness of the particles, while in [26], the focus was on adapting inertia weight to the swarm size in order to better control the search and to ensure the convergence. A fitness-oriented PSO was also proposed in [27], where authors defined speed and aggregation factors, both depending on the particles' fitnesses, then proposed to arrange the inertia weight as a moderated sum of speed and aggregation. The proposed modified PSO, m-PSO, novelty consists of proposing a fitness-based PSO which adapts the particles' behavior rather than the PSO parameters. It combines a classical inertia weight PSO with social and cognitive search mechanism to a global simplified PSO such as in [28]. In m-PSO, the swarm particles' updates are operated in different manners, some particle updates are performed according to the classical inertia weight PSO, such as in Equations (1) and (2), while some others undergo a global optimality update using Equation (4), which creates a swarm made of two sub-swarms. The particles with low-quality solutions will keep exploring the search space as a classical inertia weight PSO, while a pool of qualitative particles will search to exploit the optimality around interesting solutions around the best global solution. The proposal implements an exploration/exploitation mechanism with an iterative update where the particles may be moved from a sub-group to another according to their evolution. This process is expressed by Equations (3)-(5). For each particle, belonging to a sub-swarm depends on its fitness and on the mean of best local solutions around.