## 2 Preliminaries
### FastSLAM  
FastSLAM is a probabilistic approach that addresses the SLAM challenge by synergizing the strengths of Particle Filtering (PF) and the EKF. This method is particularly adept at handling SLAM in environments where the robot must navigate and map without prior knowledge, offering a robust solution to the problem of data association and state estimation.  
The essence of FastSLAM lies in its ability to decompose the complex SLAM problem into two parallel and conditional tasks: estimating the robot's pose and predicting the positions of landmarks within the environment. This decomposition is grounded in the principle that the prediction of landmark positions can be considered independently for each particle representing the robot's pose, thus significantly reducing the computational burden.  
In the FastSLAM framework, a set of particles is utilized to represent the distribution of the robot's pose, with each particle carrying an estimate of the trajectory and associated landmarks. The algorithm proceeds by iteratively predicting the robot's pose at each time step using a motion model, which encapsulates the control inputs and accounts for system noise. This prediction is expressed as:  
\[x_{k}=f(x_{k-1},u_{k})+\omega_{k} \tag{4}\]  
where \(x_{k}\) is the estimated pose, \(f(\cdot)\) denotes the motion model function, \(u_{k}\) represents the control inputs, and \(w_{k}\) is the process noise.  
Subsequently, the EKF is applied within the context of each particle to refine the estimates of landmark positions. This refinement is crucial for updating the map as the robot navigates through the environment. The EKF updates are based on the observation model, which correlates the robot's pose with the observed landmarks. A critical aspect of FastSLAM is the resampling step, which is triggered to maintain diversity within the particle set and to prevent the degeneration of particles that could lead to suboptimal solutions. This step ensures that particles with higher weights, indicating a better fit to the observed data, are favored, thereby enhancing the overall robustness of the algorithm.  
Despite its computational efficiency and the ability to handle large environments with numerous landmarks, FastSLAM can encounter challenges in dynamic settings where noise characteristics are not static. The algorithm's performance is contingent upon accurate noise statistics, which may vary over time. This variability can lead to inefficiencies and inaccuracies in the estimation process.  
The incorporation of multi-objective optimization strategies into the FastSLAM framework presents a compelling opportunity to further enhance its capabilities. MO-PSO's ability to handle multiple conflicting objectives simultaneously could potentially optimize various aspects of the SLAM process, such as balancing the trade-off between estimation accuracy and computational load. In this paper, by integrating MO-PSO, the FastSLAM algorithm could benefit from improved adaptability to changing environmental conditions and enhanced robustness in the presence of dynamic noise characteristics, setting the stage for more sophisticated and resilient SLAM solutions in future work.