Optimization of dynamic anti-windup is achieved for non-linear planner robot systems with actuator saturation. The static anti-windup scheme is expanded to a dynamic scheme based on passivity. The systems are represented as Euler-Lagrange systems without the gravity term, and the proportional and integral (PI) controller is used. The goal of optimization is to minimize the saturated input, i.e., the difference between the control input and the real input, using the estimation function with a semi-positive convex function of the state variable of anti-windup compensator and the saturated input. The proposed controller and its optimality are verified experimentally using a two-link robot arm.