Problem Setup

Consider a discrete-time stochastic system with full configuration $\mathfrak c_t \in \mathfrak C$, state $s_t := (\mathfrak c_t,\dot{\mathfrak c}_t)$, action $a_t \in \mathcal A$, and unknown dynamics $s_{t+1} \sim f(s_t,a_t)$. We plan from $s_0$ to a goal region using an approximate particle dynamics model $\hat f$. The prediction input is $X := (s_t,a_t)$, and its target $Y := c_{t+1} \in C$ is the task-relevant projection of the next full configuration.

Given an exchangeable calibration set $D_{\mathrm{cal}} := \{(X_i,Y_i)\}_{i=1}^{n}$, we seek an informative input-dependent region $\hat{\mathcal C}(X) \subseteq C_{\mathrm{feas}}$ with user-specified coverage. We model the feasible C-space as $C_{\mathrm{feas}} := \bigsqcup_{m\in M} S_m$ with known stratum indexer $\mathcal T:C_{\mathrm{feas}}\to M$, so regions may span full-dimensional free space and lower-dimensional contact manifolds without containing infeasible configurations.

Method: CaPTURe

CaPTURe calibrates the $k_{\mathrm{NN}}$-th-nearest-particle score within groups defined by state, action, and candidate future stratum. At inference, it queries every candidate stratum and unions the resulting feasible per-stratum regions. For a user-specified failure rate $\alpha \in (0,1)$, the union satisfies

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

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.

Aggregate planning results across six marble maze sections (180 trials per method).

Success means reaching the goal without falling into a pit. Aggregate percentages use all 180 executions per method; mean steps use successful trials only. An episode times out at step 100.

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 show CaPTURe peg-insertion rollouts across several initial peg poses.

The synchronized comparison below shows the physical execution for all six methods from the same initial peg pose.

Peg-insertion planning over 50 feasible Halton initial configurations.

Success denotes full insertion within 75 controller steps (5.0 simulated seconds). Steps are reported over successful episodes only.

This work was supported in part by the Office of Naval Research Grant N00014-24-1-2036 and NSF grants IIS-2113401 and IIS-2220876.

BibTeX (cite this!)

@misc{marques2026particlebasedconformalpredictioncontactaware,
      title={Particle-Based Conformal Prediction for Contact-Aware Uncertainty Calibration in Stratified Configuration Spaces},
      author={Luís Marques and Kristian Popov and Dmitry Berenson},
      year={2026},
      eprint={2608.09166},
      archivePrefix={arXiv},
      primaryClass={cs.RO},
      url={https://arxiv.org/abs/2608.09166},
}