  This paper derives a contact-aided inertial navigation observer for a 3D
bipedal robot using the theory of invariant observer design. Aided inertial
navigation is fundamentally a nonlinear observer design problem; thus, current
solutions are based on approximations of the system dynamics, such as an
Extended Kalman Filter (EKF), which uses a system's Jacobian linearization
along the current best estimate of its trajectory. On the basis of the theory
of invariant observer design by Barrau and Bonnabel, and in particular, the
Invariant EKF (InEKF), we show that the error dynamics of the point
contact-inertial system follows a log-linear autonomous differential equation;
hence, the observable state variables can be rendered convergent with a domain
of attraction that is independent of the system's trajectory. Due to the
log-linear form of the error dynamics, it is not necessary to perform a
nonlinear observability analysis to show that when using an Inertial
Measurement Unit (IMU) and contact sensors, the absolute position of the robot
and a rotation about the gravity vector (yaw) are unobservable. We further
augment the state of the developed InEKF with IMU biases, as the online
estimation of these parameters has a crucial impact on system performance. We
evaluate the convergence of the proposed system with the commonly used
quaternion-based EKF observer using a Monte-Carlo simulation. In addition, our
experimental evaluation using a Cassie-series bipedal robot shows that the
contact-aided InEKF provides better performance in comparison with the
quaternion-based EKF as a result of exploiting symmetries present in the system
dynamics.
