⚡ electrical engineering

Lies We Can Trust: Quantifying Action Uncertainty with Inaccurate Stochastic Dynamics through Conformalized Nonholonomic Lie groups

This paper introduces Conformal Lie-group Action Prediction Sets (CLAPS), a symmetry-aware conformal prediction algorithm that provides non-asymptotic, distribution-free probabilistic guarantees for system configurations on non-Euclidean Lie groups like SE(2), outperforming existing Euclidean-based methods in volume efficiency and uncertainty representation.

Original authors: Luís Marques, Maani Ghaffari, Dmitry Berenson

Published 2026-08-25

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Luís Marques, Maani Ghaffari, Dmitry Berenson

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Robots that move through the world, from delivery bots navigating sidewalks to drones weaving through forests, rely on mathematical models to guess where they will be a split second after they move. These models are never perfect. The real world is messy: wheels slip on wet pavement, batteries lose power, and unexpected gusts of wind push a vehicle off course. Because of these imperfections, a robot's prediction of its future location is always a guess, not a certainty. To operate safely, a robot needs to know not just where it thinks it will go, but how wide a margin of error to expect. If a robot is too confident, it might crash into a wall it thought it would miss. If it is too cautious, it might refuse to move at all, paralyzed by the fear of an unlikely mistake. The challenge for engineers is to draw a boundary around that guess—a zone that is tight enough to be useful for planning, but wide enough to guarantee that the robot will actually stay inside it, even when things go wrong.

A team of researchers at the University of Michigan has developed a new way to draw these safety boundaries, specifically for robots that move with the constraints of real-world physics, such as cars that cannot slide sideways or robots that must turn to change direction. Their method, which they call CLAPS, uses a statistical technique called conformal prediction to calibrate the robot's uncertainty. Instead of assuming the errors follow a simple, predictable pattern like a bell curve, this approach learns from a dataset of past movements to determine exactly how much the robot might deviate. The researchers found that by respecting the specific geometry of how these robots move, they could create safety zones that are significantly smaller and more accurate than those produced by existing methods. In tests with both simulated robots and a physical robot moving across a floor, their approach successfully contained the robot's true position within the predicted zone at the desired rate, while using less space than previous techniques.

The core of the problem lies in how robots represent their position. Many standard methods treat a robot's location as a simple point on a flat grid, like a dot on a piece of graph paper. This works well for simple movements, but it fails to capture the reality of robots that have complex turning and sliding behaviors. When a robot turns, its path curves in a way that a flat grid cannot easily describe without creating huge, wasteful safety zones. The researchers realized that these robots move on a curved mathematical surface, similar to how a point on a sphere moves differently than a point on a flat plane. By treating the robot's movement as happening on this curved surface, they could define the uncertainty in a way that naturally fits the robot's motion.

To test this idea, the team first built a dataset of thousands of robot movements. They took a simulated robot, the JetBot, and a real-world robot, the MBot, and commanded them to move in various ways. For every command given, they recorded where the robot actually ended up compared to where the robot's internal model predicted it would go. This collection of differences served as a calibration set. The researchers then applied their new algorithm to this data. The algorithm calculated a specific score for each movement that measured how far off the prediction was, taking into account the robot's specific turning and sliding constraints. By analyzing the distribution of these scores, the algorithm determined a scaling factor that adjusted the size of the safety zone just enough to ensure it would catch the true position of the robot with high probability.

The results showed a clear advantage for the new method. In the simulations, the researchers compared their approach against seven other common techniques. The older methods often produced safety zones that were either too large to be useful or too small to be safe. For instance, some methods created safety zones that looked like perfect circles or ellipses, which are easy to calculate but do not match the actual shape of the robot's possible errors. These shapes often included areas where the robot could never possibly go, wasting space, or missed areas where the robot was actually likely to end up. The new method, by contrast, produced safety zones that looked like curved, banana-shaped regions. These shapes followed the natural path of the robot's uncertainty, wrapping tightly around the most likely outcomes without including impossible ones.

When the researchers measured the size of these zones, the difference was significant. The new method produced safety zones that were, on average, smaller than those from the next best calibrated method. In the hardware experiments with the real MBot, the new approach created zones that were 23 percent smaller on average than the standard method, and in some cases up to 75 percent smaller. Despite being smaller, the new zones still successfully contained the robot's actual position at the rate the researchers had set as a goal. This means the robot could be more precise in its planning, allowing it to navigate tighter spaces or move faster without increasing the risk of a collision. The researchers also verified that their method worked correctly even when the robot's internal model was imperfect or when the environment introduced unexpected disturbances.

The study highlights a shift in how engineers might approach robot safety. Rather than relying on rigid assumptions about how errors behave, or using overly conservative estimates that limit a robot's capabilities, this approach uses data to learn the true shape of uncertainty. It acknowledges that the world is complex and that a robot's movement is constrained by physics in ways that simple grids cannot capture. By respecting these constraints, the researchers were able to create a tool that is both mathematically rigorous and practically efficient. The work suggests that future robots could navigate more confidently in dynamic environments, knowing exactly how much room they need to operate safely. The researchers have made their code and data available to others, inviting the broader community to build upon this method for a wide range of robotic systems.

Technical Summary: Conformal Lie-group Action Prediction Sets (CLAPS)

Problem Statement
Robotic systems operating in stochastic environments rely on imperfect dynamics models to predict future configurations. These inaccuracies stem from both aleatoric uncertainty (irreducible stochasticity like external disturbances or wheel slippage) and epistemic uncertainty (model simplifications, domain shifts, or limited data). While learning-based methods and traditional safety-critical tools exist, they often lack rigorous, distribution-free probabilistic guarantees. Traditional approaches frequently assume specific error distributions (e.g., Gaussian) or rely on uncalibrated uncertainty estimates, which are insufficient for safe control. Furthermore, existing Conformal Prediction (CP) methods, which offer non-asymptotic probabilistic guarantees, typically treat robot configurations as Euclidean vectors. This approach is suboptimal for systems with non-Euclidean configuration spaces (e.g., $SE(2)$ or $SE(3)$ ), often resulting in overly conservative, volume-inefficient prediction regions that fail to capture the inherent symmetry and geometry of the system.

Methodology
The authors propose Conformal Lie-group Action Prediction Sets (CLAPS), a symmetry-aware algorithm that constructs prediction sets guaranteed to contain the true system configuration at a user-defined probability $(1-\alpha)$ , regardless of the fidelity of the approximate dynamics model.

Theoretical Framework: The method rigorously converts nonholonomic State-Space (SS) dynamics into Lie group form. By utilizing the forced Euler-Poincaré-Suslov equations, the system's state is represented as $(g, \xi) \in G \times \mathfrak{g}$ , where $G$ is a Lie group (e.g., $SE(2)$ ) and $\mathfrak{g}$ is its Lie algebra. This allows for the propagation of uncertainty along the constraint manifold while respecting the system's inertia and nonholonomic constraints.
Symmetry-Aware Scoring: Unlike standard CP which uses Euclidean distances, CLAPS defines a nonconformity score based on the Mahalanobis distance in exponential coordinates (the Lie algebra). The score $r_1$ measures the disagreement between the predicted configuration $\tilde{g}_1$ and the true configuration $g_1$ using the group difference:
$r_1(g; \tilde{g}_1, \tilde{\Sigma}_1) = \sqrt{\log(\tilde{g}_1^{-1} g)^T \tilde{\Sigma}_1^{-1} \log(\tilde{g}_1^{-1} g)}$
Here, $\tilde{\Sigma}_1$ represents the estimated uncertainty covariance in the body-fixed frame.
Calibration and Guarantees: Using a calibration dataset of state transitions, the algorithm computes a conformal scaling factor $\zeta$ . This factor scales the approximate covariance to produce a calibrated region in the Lie algebra. Through a series of theorems, the authors prove that mapping this calibrated region back to the configuration space $Q$ via the kinematics map and group exponential results in a set $C_q$ that satisfies:
$P(q_1 \in C_q) \geq (1-\alpha)$
This guarantee holds non-asymptotically and is valid for both aleatoric and epistemic uncertainties without requiring strong assumptions about the noise distribution.
Downstream Application: The calibrated set can be used for safe control by checking if a configuration lies within the set or by reconstructing the boundary of the set in the configuration space (or workspace) for collision avoidance.

Key Contributions

Theoretical Clarification: The paper provides a concise, self-contained explanation of converting nonholonomic dynamics from State Space to Lie group form, bridging a gap in the literature.
Algorithm Development: The introduction of CLAPS, an algorithm that constructs state- and action-dependent calibrated prediction sets in $SE(2)$ . These sets provably contain the resulting configuration despite model inaccuracies and stochastic disturbances.
Symmetry-Aware Efficiency: By deriving a Lie-based, symmetry-respecting score metric, the method produces prediction regions that are smaller (more volume-efficient) than Euclidean baselines while maintaining the same probabilistic coverage.
Empirical Validation: The authors validate the approach through simulation (Isaac Sim with a JetBot) and hardware experiments (MBot), demonstrating superior volume efficiency and representation quality compared to state-of-the-art baselines.
Open Source: The implementation for both real and simulated environments is open-sourced.

Results
Experiments were conducted on a simulated JetBot (625 validation trials) and a real MBot (4,511 validation trials).

Coverage: Uncalibrated methods (SS EKF, InEKF) and data-driven estimators (InEKF+2M, InEKF+MLE) failed to meet the user-set 90% coverage requirement. All Conformal Prediction methods, including CLAPS, achieved the required marginal coverage (approx. 90%).
Volume Efficiency: CLAPS produced significantly more efficient prediction regions. In simulation, CLAPS achieved the smallest average C-space volume among all calibrated methods (normalized ratio of 1.00) and the highest Workspace Intersection-over-Union (IoU) with Monte Carlo particles (48.4%), indicating a better fit to the true uncertainty distribution.
Hardware Performance: On the MBot, CLAPS produced regions 23% smaller on average than the Euclidean CP baseline (SS EKF + CP) and up to 75% smaller in specific cases, while still satisfying the 90% coverage guarantee.
Computational Cost: The online prediction step is computationally efficient (0.02s per action), and the offline calibration is rapid (0.14s).

Significance and Claims
The paper claims that CLAPS offers a rigorous, distribution-free framework for uncertainty quantification in non-Euclidean robotic systems. By leveraging the geometric structure of Lie groups, the method overcomes the limitations of treating robot configurations as simple Euclidean points. The authors assert that this leads to "more volume-efficient prediction regions" that represent underlying uncertainty better than existing approaches. This improvement is significant for downstream tasks like safe planning and control, as tighter prediction regions allow for a larger set of probabilistically safe actions, potentially improving task speed and performance without compromising safety guarantees. The work extends previous Conformal Prediction guarantees from Euclidean spaces to robots with configurations in $SE(2)$ , addressing a critical gap in applying statistical safety guarantees to nonholonomic, stochastic robotic systems.

Technical Summary: Conformal Lie-group Action Prediction Sets (CLAPS)

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