  State estimation for legged locomotion over a dynamic rigid surface (DRS),
which is a rigid surface moving in the world frame (e.g., ships, aircraft, and
trains), remains an under-explored problem. This paper introduces an invariant
extended Kalman filter that estimates the robot's pose and velocity during DRS
locomotion by using common sensors of legged robots (e.g., inertial measurement
units (IMU), joint encoders, and RDB-D camera). A key feature of the filter
lies in that it explicitly addresses the nonstationary surface-foot contact
point and the hybrid robot behaviors. Another key feature is that, in the
absence of IMU biases, the filter satisfies the attractive group affine and
invariant observation conditions, and is thus provably convergent for the
deterministic continuous phases. The observability analysis is performed to
reveal the effects of DRS movement on the state observability, and the
convergence property of the hybrid, deterministic filter system is examined for
the observable state variables. Experiments of a Digit humanoid robot walking
on a pitching treadmill validate the effectiveness of the proposed filter under
large estimation errors and moderate DRS movement. The video of the experiments
can be found at: https://youtu.be/ScQIBFUSKzo.
