We analyze the robustness of a class of controllers that enable three-dimensional curve tracking by a free moving particle. The free particle tracks the closest point on the curve. By building a strict Lyapunov function and robustly forward invariant sets, we show input-to-state stability under predictable tolerance and safety bounds that guarantee robustness under control uncertainty, input delays, and a class of polygonal state constraints, including adaptive tracking and parameter identification under unknown control gains. Such an understanding may provide certified performance when the control laws are applied to real-life systems.