This article presents a comprehensive study on the control of linear speed and angular position in mobile robots using Proportional-Integral-Derivative (PID) controllers, with a focus on optimizing PID gains through metaheuristic optimization techniques. We begin by developing a detailed mathematical model of the robotic system, capturing its dynamics and response characteristics. Utilizing metaheuristic algorithms, such as Genetic Algorithms and Particle Swarm Optimization, we adaptively tune the PID parameters to enhance system performance in varying operational environments. The simulation results demonstrate significant improvements in trajectory tracking accuracy, reduced overshoot, and quicker settling times compared to conventional PID approaches. Additionally, the optimized PID controller showcases robust performance under different conditions, validating the effectiveness of our modeling and optimization strategy. This research not only highlights the potential of metaheuristic methods in fine-tuning PID controllers but also provides valuable insights into the simulation of mobile robotic systems, contributing to advancements in robotic control and navigation technologies.