This paper investigates the robot state estimation problem within a non-inertial environment. The proposed state estimation approach relaxes the common assumption of static ground in the system modeling. The process and measurement models explicitly treat the movement of the non-inertial environments without requiring knowledge of its motion in the inertial frame or relying on GPS or sensing environmental landmarks. Further, the proposed state estimator is formulated as an invariant extended Kalman filter (InEKF) with the deterministic part of its process model obeying the group-affine property, leading to log-linear error dynamics. The observability analysis of the filter confirms that the robot's pose (i.e., position and orientation) and velocity relative to the non-inertial environment are observable. Hardware experiments on a humanoid robot moving on a rotating and translating treadmill demonstrate the high convergence rate and accuracy of the proposed InEKF even under significant treadmill pitch sway, as well as large estimation errors.
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