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This paper reports a novel result: with proper robot models based on geometric mechanics, one can formulate the kinodynamic motion planning problems for rigid body systems as exact polynomial optimization problems. Due to the nonlinear rigid body dynamics, the motion planning problem for rigid body systems is nonconvex. Existing global optimization-based methods do not parameterize 3D rigid body motion efficiently; thus, they do not scale well to long-horizon planning problems. We use Lie groups as the configuration space and apply the variational integrator to formulate the forced rigid body dynamics as quadratic polynomials. Then, we leverage Lasserre’s hierarchy of moment relaxation to obtain the globally optimal solution via semidefinite programming. By leveraging the sparsity of the motion planning problem, the proposed algorithm has linear complexity with respect to the planning horizon. This paper demonstrates that the proposed method can provide globally optimal solutions or certificates of infeasibility at the second-order relaxation for 3D drone landing using full dynamics and inverse kinematics for serial manipulators. Moreover, we extend the algorithms to multi-body systems via the constrained variational integrators. The testing cases on cart-pole and drone with cable-suspended load suggest that the proposed algorithms can provide rank-one optimal solutions or nontrivial initial guesses. Finally, we propose strategies to speed up the computation, including an alternative formulation using quaternion, which provides empirically tight relaxations for the drone landing problem at the first-order relaxation. 

Partial point cloud registration is a challenging problem in robotics, especially when the robot undergoes a large transformation, causing a significant initial pose error and a low overlap between measurements. This letter proposes exploiting equivariant learning from 3D point clouds to improve registration robustness. We propose SE3ET, an SE(3)-equivariant registration framework that employs equivariant point convolution and equivariant transformer designs to learn expressive and robust geometric features. We tested the proposed registration method on indoor and outdoor benchmarks where the point clouds are under arbitrary transformations and lowoverlapping ratios.We also provide generalization tests and run-time performance. 

This paper reports on developing a real-time invariant proprioceptive robot state estimation framework called DRIFT. A didactic introduction to invariant Kalman filtering is provided to make this cutting-edge symmetry-preserving approach accessible to a broader range of robotics applications. Furthermore, this work dives into the development of a proprioceptive state estimation framework for dead reckoning that only consumes data from an onboard inertial measurement unit and kinematics of the robot, with two optional modules, a contact estimator and a gyro filter for low-cost robots, enabling a significant capability on a variety of robotics platforms to track the robot's state over long trajectories in the absence of perceptual data. Extensive real-world experiments using a legged robot, an indoor wheeled robot, a field robot, and a full-size vehicle, as well as simulation results with a marine robot, are provided to understand the limits of DRIFT. 

n this article, we present a novel and flexible multitask multilayer Bayesian mapping framework with readily extendable attribute layers. The proposed framework goes beyond modern metric-semantic maps to provide even richer environmental information for robots in a single mapping formalism while exploiting intralayer and interlayer correlations. It removes the need for a robot to access and process information from many separate maps when performing a complex task, advancing the way robots interact with their environments. To this end, we design a multitask deep neural network with attention mechanisms as our front-end to provide heterogeneous observations for multiple map layers simultaneously. Our back-end runs a scalable closed-form Bayesian inference with only logarithmic time complexity. We apply the framework to build a dense robotic map, including metric-semantic occupancy and traversability layers. Traversability ground truth labels are automatically generated from exteroceptive sensory data in a self-supervised manner. We present extensive experimental results on publicly available datasets and data collected by a three-dimensional bipedal robot platform and show reliable mapping performance in different environments. Finally, we also discuss how the current framework can be extended to incorporate more information, such as friction, signal strength, temperature, and physical quantity concentration using Gaussian map layers. The software for reproducing the presented results or running on customized data is made publicly available. 

Ultra-Local Models (ULM) have been applied to perform model-free control of nonlinear systems with unknown or partially known dynamics. Unfortunately, extending these methods to MIMO systems requires designing a dense input influence matrix which is challenging. This paper presents guidelines for designing an input influence matrix for discretetime, control-affine MIMO systems using an ULM-based controller. This paper analyzes the case that uses ULM and a model-free control scheme: the Hölder-continuous Finite-Time Stable (FTS) control. By comparing the ULM with the actual system dynamics, the paper describes how to extract the input-dependent part from the lumped ULM dynamics and finds that the tracking and state estimation error are coupled. The stability of the ULM-FTS error dynamics is affected by the eigenvalues of the difference (defined by matrix multiplication) between the actual and designed input influence matrix. Finally, the paper shows that a wide range of input influence matrix designs can keep the ULM-FTS error dynamics (at least locally) asymptotically stable. A numerical simulation is included to verify the result. The analysis can also be extended to other ULM-based controllers. 

This paper reports on developing an integrated framework for safety-aware informative motion planning suitable for legged robots. The information-gathering planner takes a dense stochastic map of the environment into account, while safety constraints are enforced via Control Barrier Functions (CBFs). The planner is based on the Incrementally-exploring Information Gathering (IIG) algorithm and allows closed-loop kinodynamic node expansion using a Model Predictive Control (MPC) formalism. Robotic exploration and information gathering problems are inherently path-dependent problems. That is, the information collected along a path depends on the state and observation history. As such, motion planning solely based on a modular cost does not lead to suitable plans for exploration. We propose SAFE-IIG, an integrated informative motion planning algorithm that takes into account: 1) a robot’s perceptual field of view via a submodular information function computed over a stochastic map of the environment, 2) a robot’s dynamics and safety constraints via discrete-time CBFs and MPC for closedloop multi-horizon node expansions, and 3) an automatic stopping criterion via setting an information-theoretic planning horizon. The simulation results show that SAFE-IIG can plan a safe and dynamically feasible path while exploring a dense map. 

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 extended Kalman filter (EKF) through 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 contact-aided InEKF provides better performance in comparison with the quaternion-based EKF as a result of exploiting symmetries present in system.