Surrogate models (including deep neural networks and other machine learning algorithms in supervised learning) are capable of approximating arbitrarily complex, high-dimensional input-output problems in science and engineering, but require a thorough data-agnostic uncertainty quantification analysis before these can be deployed for any safety-critical application. The standard approach for data-agnostic uncertainty quantification is to use conformal prediction (CP), a well-established framework to build uncertainty models with proven statistical guarantees that do not assume any shape for the error distribution of the surrogate model. However, since the classic statistical guarantee offered by CP is given in terms of bounds for the marginal coverage, for small calibration set sizes (which are frequent in realistic surrogate modelling that aims to quantify error at different regions), the potentially strong dispersion of the coverage distribution around its average negatively impacts the relevance of the uncertainty model's statistical guarantee, often obtaining coverages below the expected value, resulting in a less applicable framework. After providing a gentle presentation of uncertainty quantification for surrogate models for machine learning practitioners, in this paper we bridge the gap by proposing a new statistical guarantee that offers probabilistic information for the coverage of a single conformal predictor. We show that the proposed framework converges to the standard solution offered by CP for large calibration set sizes and, unlike the classic guarantee, still offers relevant information about the coverage of a conformal predictor for small data sizes. We validate the methodology in a suite of examples, and implement an open access software solution that can be used alongside common conformal prediction libraries to obtain uncertainty models that fulfil the new guarantee.

This work investigates how disturbance-aware, robustness-embedded reference trajectories translate into driving performance when executed by professional drivers in a dynamic simulator. Three planned reference trajectories are compared against a free-driving baseline (NOREF) to assess trade-offs between lap time (LT) and steering effort (SE): NOM, the nominal time-optimal trajectory; TLC, a track-limit-robust trajectory obtained by tightening margins to the track edges; and FLC, a friction-limit-robust trajectory obtained by tightening against axle and tire saturation. All trajectories share the same minimum lap-time objective with a small steering-smoothness regularizer and are evaluated by two professional drivers using a high-performance car on a virtual track. The trajectories derive from a disturbance-aware minimum-lap-time framework recently proposed by the authors, where worst-case disturbance growth is propagated over a finite horizon and used to tighten tire-friction and track-limit constraints, preserving performance while providing probabilistic safety margins. LT and SE are used as performance indicators, while RMS lateral deviation, speed error, and drift angle characterize driving style. Results show a Pareto-like LT-SE trade-off: NOM yields the shortest LT but highest SE; TLC minimizes SE at the cost of longer LT; FLC lies near the efficient frontier, substantially reducing SE relative to NOM with only a small LT increase. Removing trajectory guidance (NOREF) increases both LT and SE, confirming that reference trajectories improve pace and control efficiency. Overall, the findings highlight reference-based and disturbance-aware planning, especially FLC, as effective tools for training and for achieving fast yet stable trajectories.

We present Multi-Agent gatekeeper, a framework that provides provable safety guarantees for leader-follower formation control in cluttered 3D environments. Existing methods face a trad-off: online planners and controllers lack formal safety guarantees, while offline planners lack adaptability to changes in the number of agents or desired formation. To address this gap, we propose a hybrid architecture where a single leader tracks a pre-computed, safe trajectory, which serves as a shared trajectory backup set for all follower agents. Followers execute a nominal formation-keeping tracking controller, and are guaranteed to remain safe by always possessing a known-safe backup maneuver along the leader's path. We formally prove this method ensures collision avoidance with both static obstacles and other agents. The primary contributions are: (1) the multi-agent gatekeeper algorithm, which extends our single-agent gatekeeper framework to multi-agent systems; (2) the trajectory backup set for provably safe inter-agent coordination for leader-follower formation control; and (3) the first application of the gatekeeper framework in a 3D environment. We demonstrate our approach in a simulated 3D urban environment, where it achieved a 100% collision-avoidance success rate across 100 randomized trials, significantly outperforming baseline CBF and NMPC methods. Finally, we demonstrate the physical feasibility of the resulting trajectories on a team of quadcopters.

Safety assurance is a fundamental requirement for deploying learning-enabled autonomous systems. Hamilton-Jacobi (HJ) reachability analysis is a fundamental method for formally verifying safety and generating safe controllers. However, computing the HJ value function that characterizes the backward reachable set (BRS) of a set of user-defined failure states is computationally expensive, especially for high-dimensional systems, motivating the use of reinforcement learning approaches to approximate the value function. Unfortunately, a learned value function and its corresponding safe policy are not guaranteed to be correct. The learned value function evaluated at a given state may not be equal to the actual safety return achieved by following the learned safe policy. To address this challenge, we introduce a conformal prediction-based (CP) framework that bounds such uncertainty. We leverage CP to provide probabilistic safety guarantees when using learned HJ value functions and policies to prevent control systems from reaching failure states. Specifically, we use CP to calibrate the switching between the unsafe nominal controller and the learned HJ-based safe policy and to derive safety guarantees under this switched policy. We also investigate using an ensemble of independently trained HJ value functions as a safety filter and compare this ensemble approach to using individual value functions alone.

Dynamical system (DS)-based learning from demonstration (LfD) is a powerful tool for generating motion plans in the operation (`task') space of robotic systems. However, the realization of the generated motion plans is often compromised by a ''task-execution mismatch'', where unmodeled dynamics, persistent disturbances, and system latency cause the robot's actual task-space state to diverge from the desired motion trajectory. We propose a novel task-level robust control architecture, L1-augmented Dynamical Systems (L1-DS), that explicitly handles the task-execution mismatch in tracking a nominal motion plan generated by any DS-based LfD scheme. Our framework augments any DS-based LfD model with a nominal stabilizing controller and an L1 adaptive controller. Furthermore, we introduce a windowed Dynamic Time Warping (DTW)-based target selector, which enables the nominal stabilizing controller to handle temporal misalignment for improved phase-consistent tracking. We demonstrate the efficacy of our architecture on the LASA and IROS handwriting datasets.

Robotic systems navigating in real-world settings require a semantic understanding of their environment to properly determine safe actions. This work aims to develop the mathematical underpinnings of such a representation -- specifically, the goal is to develop safety filters that are risk-aware. To this end, we take a two step approach: encoding an understanding of the environment via Poisson's equation, and associated risk via Laplace guidance fields. That is, we first solve a Dirichlet problem for Poisson's equation to generate a safety function that encodes system safety as its 0-superlevel set. We then separately solve a Dirichlet problem for Laplace's equation to synthesize a safe \textit{guidance field} that encodes variable levels of caution around obstacles -- by enforcing a tunable flux boundary condition. The safety function and guidance fields are then combined to define a safety constraint and used to synthesize a risk-aware safety filter which, given a semantic understanding of an environment with associated risk levels of environmental features, guarantees safety while prioritizing avoidance of higher risk obstacles. We demonstrate this method in simulation and discuss how \textit{a priori} understandings of obstacle risk can be directly incorporated into the safety filter to generate safe behaviors that are risk-aware.

When do diffusion models reproduce their training data, and when are they able to generate samples beyond it? A practically relevant theoretical understanding of this interplay between memorization and generalization may significantly impact real-world deployments of diffusion models with respect to issues such as copyright infringement and data privacy. In this work, to disentangle the different factors that influence memorization and generalization in practical diffusion models, we introduce a scientific and mathematical "laboratory" for investigating these phenomena in diffusion models trained on fully synthetic or natural image-like structured data. Within this setting, we hypothesize that the memorization or generalization behavior of an underparameterized trained model is determined by the difference in training loss between an associated memorizing model and a generalizing model. To probe this hypothesis, we theoretically characterize a crossover point wherein the weighted training loss of a fully generalizing model becomes greater than that of an underparameterized memorizing model at a critical value of model (under)parameterization. We then demonstrate via carefully-designed experiments that the location of this crossover predicts a phase transition in diffusion models trained via gradient descent, validating our hypothesis. Ultimately, our theory enables us to analytically predict the model size at which memorization becomes predominant. Our work provides an analytically tractable and practically meaningful setting for future theoretical and empirical investigations. Code for our experiments is available at https://github.com/DruvPai/diffusion_mem_gen.

With multi-agent systems increasingly deployed autonomously at scale in complex environments, ensuring safety of the data-driven policies is critical. Control Barrier Functions have emerged as an effective tool for enforcing safety constraints, yet existing learning-based methods often lack in scalability, generalization and sampling efficiency as they overlook inherent geometric structures of the system. To address this gap, we introduce symmetries-infused distributed Control Barrier Functions, enforcing the satisfaction of intrinsic symmetries on learnable graph-based safety certificates. We theoretically motivate the need for equivariant parametrization of CBFs and policies, and propose a simple, yet efficient and adaptable methodology for constructing such equivariant group-modular networks via the compatible group actions. This approach encodes safety constraints in a distributed data-efficient manner, enabling zero-shot generalization to larger and denser swarms. Through extensive simulations on multi-robot navigation tasks, we demonstrate that our method outperforms state-of-the-art baselines in terms of safety, scalability, and task success rates, highlighting the importance of embedding symmetries in safe distributed neural policies.

As prominent real-time safety-critical reactive control techniques, Control Barrier Function Quadratic Programs (CBF-QPs) work for control affine systems in general but result in local minima in the generated trajectories and consequently cannot ensure convergence to the goals. Contrarily, Modulation of Dynamical Systems (Mod-DSs), including normal, reference, and on-manifold Mod-DS, achieve obstacle avoidance with few and even no local minima but have trouble optimally minimizing the difference between the constrained and the unconstrained controller outputs, and its applications are limited to fully-actuated systems. We dive into the theoretical foundations of CBF-QP and Mod-DS, proving that despite their distinct origins, normal Mod-DS is a special case of CBF-QP, and reference Mod-DS's solutions are mathematically connected to that of the CBF-QP through one equation. Building on top of the unveiled theoretical connections between CBF-QP and Mod-DS, reference Mod-based CBF-QP and on-manifold Mod-based CBF-QP controllers are proposed to combine the strength of CBF-QP and Mod-DS approaches and realize local-minimum-free reactive obstacle avoidance for control affine systems in general. We validate our methods in both simulated hospital environments and real-world experiments using Ridgeback for fully-actuated systems and Fetch robots for underactuated systems. Mod-based CBF-QPs outperform CBF-QPs as well as the optimally constrained-enforcing Mod-DS approaches we proposed in all experiments.

The widespread distribution of microplastics (MPs) in the environment presents significant challenges for their detection and identification. Fluorescence imaging has emerged as a promising technique for enhancing plastic particle detectability and enabling accurate classification based on fluorescence behavior. However, conventional segmentation techniques face limitations, including poor signal-to-noise ratio, inconsistent illumination, thresholding difficulties, and false positives from natural organic matter (NOM). To address these challenges, this study introduces the Fluorescence Imaging Microplastic Analysis Platform (FIMAP), a retrofitted multispectral camera with four optical filters and five excitation wavelengths. FIMAP enables comprehensive characterization of the fluorescence behavior of ten Nile Red-stained MPs: HDPE, LDPE, PP, PS, EPS, ABS, PVC, PC, PET, and PA, while effectively excluding NOM. Using K-means clustering for robust segmentation (Intersection over Union = 0.877) and a 20-dimensional color coordinate multivariate nearest neighbor approach for MP classification (>3.14 mm), FIMAP achieves 90% precision, 90% accuracy, 100% recall, and an F1 score of 94.7%. Only PS was occasionally misclassified as EPS. For smaller MPs (35-104 microns), classification accuracy declined, likely due to reduced stain sorption, fewer detectable pixels, and camera instability. Integrating FIMAP with higher-magnification instruments, such as a microscope, may enhance MP identification. This study presents FIMAP as an automated, high-throughput framework for detecting and classifying MPs across large environmental sample volumes.