Legged robots require knowledge of pose and velocity in order to maintain stability and execute walking paths. Current solutions either rely on vision data, which is susceptible to environmental and lighting conditions, or fusion of kinematic and contact data with measurements from an inertial measurement unit (IMU). In this work, we develop a contact-aided invariant extended Kalman filter (InEKF) using the theory of Lie groups and invariant observer design. This filter combines contact-inertial dynamics with forward kinematic corrections to estimate pose and velocity along with all current contact points. We show that the error dynamics follows a log-linear autonomous differential equation with several important consequences: (a) the observable state variables can be rendered convergent with a domain of attraction that is independent of the system's trajectory; (b) unlike the standard EKF, neither the linearized error dynamics nor the linearized observation model depend on the current state estimate, which (c) leads to improved convergence properties and (d) a local observability matrix that is consistent with the underlying nonlinear system. Furthermore, we demonstrate how to include IMU biases, add/remove contacts, and formulate both world-centric and robo-centric versions. We compare the convergence of the proposed InEKF with the commonly used quaternion-based EKF though both simulations and experiments on a Cassie-series bipedal robot. Filter accuracy is analyzed using motion capture, while a LiDAR mapping experiment provides a practical use case. Overall, the developed contactaided InEKF provides better performance in comparison with the quaternion-based EKF as a result of exploiting symmetries present in system. changes in lighting as well as the operating environment. It is therefore beneficial to develop a low-level state estimator that fuses data only from proprioceptive sensors to form accurate high-frequency state estimates. This approach was taken by Bloesch et al. [15] when developing a quaternion-based extended Kalman filter (QEKF) that combines inertial, contact, and kinematic data to estimate the robot's base pose, velocity, and a number of contact states. In this article, we expand upon these ideas to develop an invariant extended Kalman filter (InEKF) that has improved convergence and consistency properties allowing for a more robust observer that is suitable for long-term autonomy.The theory of invariant observer design is based on the estimation error being invariant under the action of a matrix Lie group [1,20], which has recently led to the development of the InEKF 1 [18,8,10,11] with successful applications and promising results in simultaneous localization and mapping [8,86] and aided inertial navigation systems [4,5,8,83]. The invariance of the estimation error with respect to a Lie group action is referred to as a symmetry of the system [5]. Summarized briefly, Barrau and Bonnabel [10] showed that if the state is defined on a Lie group, and the dynamics satisfy a particular "group affin...

This paper develops a Bayesian continuous 3D semantic occupancy map from noisy point cloud measurements. In particular, we generalize the Bayesian kernel inference model for occupancy (binary) map building to semantic (multi-class) maps. The method nicely reverts to the original occupancy mapping framework when only one occupied class exists in obtained measurements. First, using Categorical likelihood and its conjugate prior distribution, we extend the counting sensor model for binary classification to a multi-class classification problem which results in a unified probabilistic model for both occupancy and semantic probabilities. Secondly, by applying a Bayesian spatial kernel inference to the semantic counting sensor model, we relax the independent grid assumption and bring smoothness and continuity to the map inference. These latter properties enable the method to exploit local correlations present in the environment to predict semantic probabilities in regions unobserved by the sensor while increasing the performance. Lastly, computational efficiency and scalability are achieved by leveraging sparse kernels and a test-data octrees data structure. The evaluations using multiple sequences of stereo camera and LiDAR datasets show that the proposed method consistently outperforms the compared baselines. We also present a qualitative evaluation using data collected by a biped robot platform on the University of Michigan -North Campus.

Abstract-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 quaternionbased 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.

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