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It is an important ability for any mobile robot to be able to estimate its posture and to gauge the distance it traveled. In this paper, we have addressed this problem in a dynamic quadruped robot by combining traditional state estimation methods with machine learning. We have designed and implemented a navigation algorithm for full body state (position, velocity, and attitude) estimation that uses no external reference but relies on multimodal proprioceptive sensory information only. The extended Kalman filter (EKF) was used to provide error estimation and data fusion from two independent sources of information: 1) strapdown mechanization algorithm processing raw inertial data and 2) legged odometry. We have devised a novel legged odometer that combines information from a multimodal combination of sensors (joint and pressure). We have shown our method to work for a dynamic turning gait, and we have also successfully demonstrated how it generalizes to different velocities and terrains. Furthermore, our solution proved to be immune to substantial slippage of the robot's feet.