Abstract Reliable uncertainty representation is essential for deploying autonomous systems that interact with their environment, as robots must reason about how uncertainty arising from both stochasticity and model mismatch is impacted by contacts with obstacles (e.g., when navigating through a cluttered environment or inserting a part into an assembly). We propose Calibrated Particle-sets for Trans-dimensional Uncertainty Representation (CaPTURe), a geometry-aware, conformal prediction-based algorithm that generates probabilistically-valid prediction regions of the unknown future system configuration using particle-based models of arbitrary fidelity. While calibrated uncertainty predictions are essential for safe and efficient planning, analytical or learned motion models are often inaccurate - due to limited data, simplifying assumptions, unmodeled effects, etc. - which can lead to unsafe executions or task failure. Additionally, when a robot contacts an obstacle, the distribution of its future configurations can become multi-modal, disjoint, and can lie along manifolds of lower intrinsic dimension than the space of possible robot configurations. Our method uses a calibration dataset of system transitions to locally calibrate motion uncertainty estimates, constructing regions guaranteed to contain the future robot configuration at a user-set likelihood. Our calibration procedure captures how motion uncertainty varies between contact-rich and contactless motions, leading to sufficient coverage in both cases. We evaluate our method on two simulated planning tasks: controlling a marble around a labyrinth and tight-tolerance peg-in-hole insertion by a manipulator in simulation. Compared to relevant baselines, CaPTURe achieves the user-specified coverage requirement, when both in and out of contact, and leads to executions that are up to 37% more successful.

Problem Setup

Consider a discrete-time stochastic system with configuration $\mathfrak c_t \in \mathfrak C$ , state $s_t := (\mathfrak c_t,\dot{\mathfrak c}_t)$ , action $u_t \in \mathcal U$ , and unknown dynamics $s_{t+1} \sim f(s_t,u_t)$ . We consider the problem of moving this robot, safely and efficiently, from an initial state $s_0$ to a goal region in a known environment using an approximate particle-based dynamics model $\hat f$ of arbitrary fidelity. During planning, we can use the current state and action $X := (s_t,u_t) = (\mathfrak c_t,\dot{\mathfrak c}_t,u_t)$ to predict the future configuration $Y := c_{t+1}$ . Yet, $\hat f$ ’s predictions can be unreliable due to aleatoric disturbances, epistemic model mismatch, and contact-induced changes in feasible motion. To support uncertainty-aware planning and control under contact, we construct calibrated prediction regions over $C \subseteq \mathfrak C$ a task-relevant subset of the configuration space — $c_t \subseteq \mathfrak c_t$ denotes the corresponding task-relevant robot configurations.

Given a finite exchangeable calibration set $D_{\mathrm{cal}} := \{(X_i,Y_i)\}_{i=1}^{n}$ , our aim is to construct an input-dependent prediction region $\hat{\mathcal C} \subseteq C$ that is calibrated, informative for planning, and constrained to feasible configurations – $\hat{\mathcal C}(X) \cap \mathcal C_{\mathrm{infeasible}} = \emptyset$ . We model $C$ as a finite stratified space $C := \bigsqcup_{m\in M} S_m$ with known stratum indexer $\mathcal T:C \to \{1,\ldots,|M|\}$ , allowing feasible configurations to include free-space regions as well as lower-dimensional contact manifolds, such as wall-contact edges and corner contacts.

Method: CaPTURe

Given a user-specified acceptable failure-rate $\alpha \in (0,1)$ , CaPTURe constructs a state-action-stratum dependent prediction region $\hat{\mathcal C} \subseteq C$ , such that

$\mathbb P\!\left(c_{t+1} \in \hat{\mathcal C}\right) \ge 1 - \alpha.$

The resulting prediction region $\hat{\mathcal C} = \bigcup_{S_m \in \mathcal S} \hat{\mathcal C}_{S_m}(s_t,u_t,S_m) \subseteq C$ is a union of regions built over the robot’s configuration strata.

Offline calibration diagram — **Offline calibration of CaPTURe.** For each transition in $D_{\mathrm{cal}}$ , $\hat f$ receives the current full-system state $s_t := (\mathfrak{c}_t,\dot{\mathfrak{c}}_t)$ and action $u_t$ to sample $L$ predictive particles (black) of the future target configuration $c_{t+1}$ which we use to compute the nonconformity score $R_i$ (blue). A subset of transition-scores $D_{\mathrm{cal}}^{part}$ is used to fit a regression decision-tree that separates the input space – state, velocity and future strata $S(c_{t+1})$ – into groups with approximately group-constant prediction score. The holdout subset $D_{\mathrm{cal}}^{cp}$ is then passed through the `DTree` with each example landing in a leaf node and hence corresponding group $k$ . SplitCP is performed independently for each group, resulting in a per-partition conformal threshold $\hat q_k$ .

Prediction region construction diagram — **Construction of stratified prediction region $\hat{\mathcal C}$ .** Given an action $u_t$ and current full-system state $s_t := (\mathfrak{c}_t,\dot{\mathfrak{c}}_t)$ the predictive model $\hat f$ returns predictive particles (black) representing possible future target configurations. For each candidate future stratum $(S_m)$ in C-space, we query the pre-fit `DTree` with $(s_t,u_t,S_m)$ to get a contact-aware uncertainty threshold $\hat q_k$ . For each stratum, we evaluate which configurations in $S_m$ have score $\le \hat q_k$ , where the score is the $k$ -th distance between a candidate configuration in $S_m$ and the particles. Finally, we compose the full prediction region $\hat{\mathcal C}(s_t,u_t,\mathcal S)$ as the union of the per-stratum regions over candidate future strata $S_m \in \mathcal S$ .

Experiments

Marble Labyrinth Control

We simulate a planar marble control environment inspired by the BRIO labyrinth toy, where the task is to navigate a tight-clearance maze while avoiding known pit locations under aleatoric disturbances and significant model mismatch. The state is $s_t=[x^b,\dot x^b,y^b,\dot y^b,\alpha,\beta]^\top$ , with board-fixed marble coordinates and plate inclination angles, and controls are motor velocities that tilt the board. We construct prediction regions over the 2D marble position $c_t = [x^b, y^b]^\top$ . This task is challenging since the controls only indirectly influence marble location through plate tilt, causing delayed responses, momentum accumulation, and wall-shaped uncertainty when the dynamics are inaccurately modeled. The videos below compare rollouts produced by CaPTURe and the baselines across maze sections.

Map

Episode

Method

Map

Manipulator Peg Insertion

We further evaluate CaPTURe on a tight-tolerance peg insertion task adapted from the Factory simulation suite in Isaac Sim. We control a Franka Panda (7 DoF manipulator) to insert a cylindrical peg into a low-clearance hole under both stochastic disturbances and significant model mismatch. To facilitate contact-aware planning, we restrict end-effector motion to lie along the hole's plane, reducing possible peg poses from $SE(3)$ to $SE(2)$ and making the configuration of interest $c_t = [x,z,\theta]$ . The videos below compare single-episode and all-method rollouts across initial peg poses.

Initial peg pose