This paper develops a high-performance switched nonlinear PID controller for trajectory tracking of robot manipulators in the presence of bounded disturbances. The proposed control law is based on an independent nonlinear PID architecture that incorporates a non-singular switching gain, smooth saturation nonlinearity, and a hierarchical composite error, resulting in a computationally efficient and fully model-free implementation. Rigorous Lyapunov analysis establishes global practical stability and practical exponential convergence of the tracking error to a tunable residual set using only known upper bounds of the manipulator dynamics (inertia, Coriolis/centrifugal, and gravity terms). The control effort is analytically characterized, showing that although the torque may grow linearly during transients, it remains explicitly bounded in steady state with user-tunable limits compatible with actuator constraints. All controller parameters are systematically optimized offline using the Butterfly Optimization Algorithm (BOA) within bounds informed by their functional definitions, thereby eliminating heuristic gain tuning. Numerical simulations on a standard 2-DOF robotic manipulator validate the theoretical guarantees and demonstrate improved tracking accuracy and disturbance robustness compared with conventional PID-type approaches.