  This paper proposes a state estimator for legged robots operating in slippery
environments. An Invariant Extended Kalman Filter (InEKF) is implemented to
fuse inertial and velocity measurements from a tracking camera and leg
kinematic constraints. {\color{black}The misalignment between the camera and
the robot-frame is also modeled thus enabling auto-calibration of camera pose.}
The leg kinematics based velocity measurement is formulated as a
right-invariant observation. Nonlinear observability analysis shows that other
than the rotation around the gravity vector and the absolute position, all
states are observable except for some singular cases. Discrete observability
analysis demonstrates that our filter is consistent with the underlying
nonlinear system. An online noise parameter tuning method is developed to adapt
to the highly time-varying camera measurement noise. The proposed method is
experimentally validated on a Cassie bipedal robot walking over slippery
terrain. A video for the experiment can be found at
https://youtu.be/VIqJL0cUr7s.
