Quadruped robots must exhibit robust walking capabilities in practical applications. In this work, we propose a novel approach that enables quadruped robots to pass various small obstacles, or "tiny traps". Existing methods often rely on exteroceptive sensors, which can be unreliable for detecting such tiny traps. To overcome this limitation, our approach focuses solely on proprioceptive inputs. We introduce a two-stage training framework incorporating a contact encoder and a classification head to learn implicit representations of different traps. Additionally, we design a set of tailored reward functions to improve both the stability of training and the ease of deployment for goal-tracking tasks. To benefit further research, we design a new benchmark for tiny trap task. Extensive experiments in both simulation and real-world settings demonstrate the effectiveness and robustness of our method. Project Page: https://robust-robot-walker.github.io/