We propose Conformal Lie-group Action Prediction Sets (CLAPS), a symmetry-aware conformal prediction-based algorithm that constructs, for a given action, a set guaranteed to contain the resulting system configuration at a user-defined probability. Our assurance holds under both aleatoric and epistemic uncertainty, non-asymptotically, and does not require strong assumptions about the true system dynamics, the uncertainty sources, or the quality of the approximate dynamics model. Typically, uncertainty quantification is tackled by making strong assumptions about the error distribution or magnitude, or by relying on uncalibrated uncertainty estimates — i.e., with no link to frequentist probabilities — which are insufficient for safe control. Recently, conformal prediction has emerged as a statistical framework capable of providing distribution-free probabilistic guarantees on test-time prediction accuracy. While current conformal methods treat robots as Euclidean points, many systems have non-Euclidean configurations, e.g., some mobile robots have $SE(2)$. In this work, we rigorously analyze configuration errors using Lie groups, extending previous Euclidean Space theoretical guarantees to $SE(2)$. Our experiments on a simulated Jetbot, and on a real MBot, suggest that by considering the configuration space’s structure, our symmetry-informed nonconformity score leads to more volume-efficient prediction regions that represent the underlying uncertainty better than existing approaches.

CLAPS

CLAPS uses a dataset of state transitions $(D_{cal})$ to calibrate the uncertainty estimates provided by approximate dynamics models. CLAPS can be applied as a post-hoc calibration layer on top of existing Lie-algebraic Gaussian uncertainty estimators (e.g., Invariant EKF), turning their approximate covariances into provably calibrated ones. By using a symmetry-respective score metric, our approach produces prediction regions that are more volume-efficient than existing conformal prediction baselines that treat the robot’s configuration as Euclidean.

The prediction region constructed by CLAPS $(C^q \subseteq Q)$ can be used for probably-safe control in three main ways (for more details refer to Section $\S$V-C):

1. Configuration Check: a (sample) configuration $g$ belongs in $C^q$ if $\sqrt{\log(\tilde{g}^{-1}g)^\top \tilde{\Sigma}^{-1}\log(\tilde{g}^{-1}g)} \le \chi^2_{\alpha}(\dim \mathfrak g)$ — quick to evaluate in batch
2. C-space set: The $C^q$ can be reconstructed by Alg. 2, for example to check if $C^q \subseteq \mathcal Q_{safe}$, for a known safe set $\mathcal Q_{safe} \subseteq \mathcal Q$.
3. Workspace set: $C^q$ can be inflated by the robot’s radius and mapped to the workspace $(\mathbb R^2)$ to perform collision checks with known obstacles.

Experiments

We compare CLAPS against seven baselines in both simulation (JetBot) and hardware (MBot) to demonstrate its improved efficiency and representation quality. We model both systems as a second-order unicycles, and perform standard system identification to estimate the inertial properties. In all the experiments below we use $\alpha=0.1$.

A) JetBot Experiments (Simulation)
 In Isaac Sim, we independently sampled additive perturbations to $u_{des}$, introducing aleatoric uncertainty. This leads to the well-known banana-shaped distributions seen below. Epistemic uncertainty arose from unmodeled effects (e.g., friction), and imperfections in the mass/inertia estimation. The Figure below demonstrates CLAPS’ ability to represent the underlying dynamics uncertainty of the unknown system (MC particles).

Quantitatively, CLAPS achieves the highest average Intersection over Union (IoU) with the MC particles, validating its alignment with the systems’ uncertainty propagation, and CLAPS has a smaller C-space volume than all calibrated baselines in each of the 625 validation trials we tested.

Below we visualize the C-space regions $C^q$ constructed by the different methods in three of the 625 validation trials. The State Space (SS) baselines produce hyperellipsoids in configuration space, due to treating it as Euclidean. Instead, both the Invariant Kalman Filter (InEKF) and CLAPS produce symmetry-respective prediction regions, better capturing the underlying uncertainty. While the uncertainty estimates provided by the InEKF are approximate, CLAPS provides provably calibrated prediction regions suitable for safe-control.

B) MBot Experiments (Hardware)
 We also validated our method on an MBot, a differential-drive vehicle shown below. Despite a relatively-small calibration dataset corresponding to $\approx$2 min of driving data $(\lvert D_{cal}\rvert = 237)$, our method provably satisfied the user-specified safety specifications, thanks to its non-asymptotic guarantees. CLAPS uses $D_{cal}$ to derive data-driven provable (probabilistic) bounds on the uncertainty arising from both model mismatch, and inherent stochasticity.

The system configuration and velocity were estimated using a motion capture system. Uncertainty in the resulting configuration arose due to inaccuracies in inertial property estimation, actuation delays, center-of-mass deviation from the body-fixed origin, ground-surface imperfections, friction, network jitter, etc. The collection procedure of system transitions that make up $D_{cal}$ and the validation set is shown below.

Our Python-implementation of CLAPS can run at 25 Hz, the sampling frequency of the MBot’s sensors, making it serviceable for online use.

BibTeX (cite this!)

@misc{marques2025liestrustquantifyingaction,
      title={Lies We Can Trust: Quantifying Action Uncertainty with Inaccurate Stochastic Dynamics through Conformalized Nonholonomic Lie Groups}, 
      author={Luís Marques and Maani Ghaffari and Dmitry Berenson},
      year={2025},
      eprint={2512.10294},
      archivePrefix={arXiv},
      primaryClass={cs.RO},
      url={https://arxiv.org/abs/2512.10294}, 
}