Risk-aware MPPI for Stochastic Hybrid Systems

Path Planning for stochastic hybrid systems presents a unique challenge of predicting distributions of future states subject to a state-dependent dynamics switching function. In this work, we propose a variant of Model Predictive Path Integral Control (MPPI) to plan kinodynamic paths for such systems. Monte Carlo may be inaccurate when few samples are chosen to predict future states under state-dependent disturbances. We employ recently proposed Unscented Transform-based methods to capture stochasticity in the states as well as the state-dependent switching surfaces. This is in contrast to previous works that perform switching based only on the mean of predicted states. We focus our motion planning application on the navigation of a mobile robot in the presence of dynamically moving agents whose responses are based on sensor-constrained attention zones. We evaluate our framework on a simulated mobile robot and show faster convergence to a goal without collisions when the robot exploits the hybrid human dynamics versus when it does not.

RPCBF: Constructing Safety Filters Robust to Model Error and Disturbances via Policy Control Barrier Functions

Control Barrier Functions (CBFs) have proven to be an effective tool for performing safe control synthesis for nonlinear systems. However, guaranteeing safety in the presence of disturbances and input constraints for high relative degree systems is a difficult problem. In this work, we propose the Robust Policy CBF (RPCBF), a practical method of constructing CBF approximations that is easy to implement and robust to disturbances via the estimation of a value function. We demonstrate the effectiveness of our method in simulation on a variety of high relative degree input-constrained systems. Finally, we demonstrate the benefits of RPCBF in compensating for model errors on a hardware quadcopter platform by treating the model errors as disturbances. The project page can be found at https://oswinso.xyz/rpcbf.

Model Predictive Path Integral Methods with Reach-Avoid Tasks and Control Barrier Functions

The rapid advancement of robotics necessitates robust tools for developing and testing safe control architectures in dynamic and uncertain environments. Ensuring safety and reliability in robotics, especially in safety-critical applications, is crucial, driving substantial industrial and academic efforts. In this context, we extend CBFkit, a Python/ROS2 toolbox, which now incorporates a planner using reach-avoid specifications as a cost function. This integration with the Model Predictive Path Integral (MPPI) controllers enables the toolbox to satisfy complex tasks while ensuring formal safety guarantees under various sources of uncertainty using Control Barrier Functions (CBFs). CBFkit is optimized for speed using JAX for automatic differentiation and jaxopt for quadratic program solving. The toolbox supports various robotic applications, including autonomous navigation, human-robot interaction, and multi-robot coordination. The toolbox also offers a comprehensive library of planner, controller, sensor, and estimator implementations. Through a series of examples, we demonstrate the enhanced capabilities of CBFkit in different robotic scenarios.

Biharmonic Distance of Graphs and its Higher-Order Variants: Theoretical Properties with Applications to Centrality and Clustering

Effective resistance is a distance between vertices of a graph that is both theoretically interesting and useful in applications. We study a variant of effective resistance called the biharmonic distance. While the effective resistance measures how well-connected two vertices are, we prove several theoretical results supporting the idea that the biharmonic distance measures how important an edge is to the global topology of the graph. Our theoretical results connect the biharmonic distance to well-known measures of connectivity of a graph like its total resistance and sparsity. Based on these results, we introduce two clustering algorithms using the biharmonic distance. Finally, we introduce a further generalization of the biharmonic distance that we call the $k$-harmonic distance. We empirically study the utility of biharmonic and $k$-harmonic distance for edge centrality and graph clustering.

CBFKIT: A Control Barrier Function Toolbox for Robotics Applications

This paper introduces CBFKit, a Python/ROS toolbox for safe robotics planning and control under uncertainty. The toolbox provides a general framework for designing control barrier functions for mobility systems within both deterministic and stochastic environments. It can be connected to the ROS open-source robotics middleware, allowing for the setup of multi-robot applications, encoding of environments and maps, and integrations with predictive motion planning algorithms. Additionally, it offers multiple CBF variations and algorithms for robot control. The CBFKit is demonstrated on the Toyota Human Support Robot (HSR) in both simulation and in physical experiments.

Comparing Graph Transformers via Positional Encodings

The distinguishing power of graph transformers is closely tied to the choice of positional encoding: features used to augment the base transformer with information about the graph. There are two primary types of positional encoding: absolute positional encodings (APEs) and relative positional encodings (RPEs). APEs assign features to each node and are given as input to the transformer. RPEs instead assign a feature to each pair of nodes, e.g., graph distance, and are used to augment the attention block. A priori, it is unclear which method is better for maximizing the power of the resulting graph transformer. In this paper, we aim to understand the relationship between these different types of positional encodings. Interestingly, we show that graph transformers using APEs and RPEs are equivalent in terms of distinguishing power. In particular, we demonstrate how to interchange APEs and RPEs while maintaining their distinguishing power in terms of graph transformers. Based on our theoretical results, we provide a study on several APEs and RPEs (including the resistance distance and the recently introduced stable and expressive positional encoding (SPE)) and compare their distinguishing power in terms of transformers. We believe our work will help navigate the huge number of choices of positional encoding and will provide guidance on the future design of positional encodings for graph transformers.

Feasible Space Monitoring for Multiple Control Barrier Functions with application to Large Scale Indoor Navigation

Quadratic programs (QP) subject to multiple time-dependent control barrier function (CBF) based constraints have been used to design safety-critical controllers. However, ensuring the existence of a solution at all times to the QP subject to multiple CBF constraints is non-trivial. We quantify the feasible solution space of the QP in terms of its volume. We introduce a novel feasible space volume monitoring control barrier function that promotes compatibility of barrier functions and, hence, existence of a solution at all times. We show empirically that our approach not only enhances feasibility but also exhibits reduced sensitivity to changes in the hyperparameters such as gains of nominal controller. Finally, paired with a global planner, we evaluate our controller for navigation among humans in the AWS Hospital gazebo environment. The proposed controller is demonstrated to outperform the standard CBF-QP controller in maintaining feasibility.

Safe Control Synthesis for Hybrid Systems through Local Control Barrier Functions

Control Barrier Functions (CBF) have provided a very versatile framework for the synthesis of safe control architectures for a wide class of nonlinear dynamical systems. Typically, CBF-based synthesis approaches apply to systems that exhibit nonlinear -- but smooth -- relationship in the state of the system and linear relationship in the control input. In contrast, the problem of safe control synthesis using CBF for hybrid dynamical systems, i.e., systems which have a discontinuous relationship in the system state, remains largely unexplored. In this work, we build upon the progress on CBF-based control to formulate a theory for safe control synthesis for hybrid dynamical systems. Under the assumption that local CBFs can be synthesized for each mode of operation of the hybrid system, we show how to construct CBF that can guarantee safe switching between modes. The end result is a switching CBF-based controller which provides global safety guarantees. The effectiveness of our proposed approach is demonstrated on two simulation studies.

Understanding Oversquashing in GNNs through the Lens of Effective Resistance

Message passing graph neural networks are popular learning architectures for graph-structured data. However, it can be challenging for them to capture long range interactions in graphs. One of the potential reasons is the so-called oversquashing problem, first termed in [Alon and Yahav, 2020], that has recently received significant attention. In this paper, we analyze the oversquashing problem through the lens of effective resistance between nodes in the input graphs. The concept of effective resistance intuitively captures the "strength" of connection between two nodes by paths in the graph, and has a rich literature connecting spectral graph theory and circuit networks theory. We propose the use the concept of total effective resistance as a measure to quantify the total amount of oversquashing in a graph, and provide theoretical justification of its use. We further develop algorithms to identify edges to be added to an input graph so as to minimize the total effective resistance, thereby alleviating the oversquashing problem when using GNNs. We provide empirical evidence of the effectiveness of our total effective resistance based rewiring strategies.