To address the problem of rapid decline in positioning accuracy for quadruped robots when satellite signals are unavailable and environmental perception degrades, this paper proposes an inertial navigation algorithm for quadruped robots assisted by foot-end inertial information. Firstly, a leg odometry observation model is constructed based on foot-end inertial data and joint encoder readings to compensate for velocity loss caused by the stationary contact assumption. Subsequently, a temporal convolutional network (TCN) and bidirectional gated recurrent unit (BiGRU) are employed to extract long-term and short-term features from the footend inertial and joint data, enabling robust estimation of stationary contact intervals. The proposed odometry observation model is employed as measurement input for an Invariant Extended Kalman Filter (InEKF) to correct inertial navigation errors during stationary intervals. Long-distance outdoor localization experiments demonstrated the effectiveness of the algorithm, achieving over 96% accuracy in stationary interval estimation. In open-loop tests, the endpoint error was only 0.93% of the total traveled distance. In mixed-terrain closed-loop experiments, the average eastward and northward errors were 1.07 m and 0.74 m, respectively, highlighting the proposed method's ability to maintain high positioning accuracy over extended periods without relying on external data.