Control-Ready Uncertainty for Trajectory Diffusion
SCOPE — Score Curvature for Online Precision Estimation

Authors: Zhiwei Xue, Jia Yue Kam, Jinhang Qiu, Yifeng Cheng, Ege Gursoy, Jiaming Wang, Vincent Bonnet, Harold Soh

Affiliations: National University of Singapore. Smart Systems Institute, National University of Singapore. LAAS-CNRS

Abstract
Diffusion models can represent complex, multimodal trajectory distributions, but extracting uncertainty from them typically requires costly Monte Carlo sampling. This limits their use in real-time control, where robots must rapidly assess risk and maintain safety margins. We introduce Score-Curvature for Online Precision Estimation (SCOPE), a lightweight module that augments diffusion trajectory models with control-ready uncertainty. SCOPE learns a structured precision matrix around each nominal trajectory by distilling score-curvature information and producing calibrated Gaussian tubes with low overhead and without repeated Monte Carlo sampling. These tubes provide per-timestep covariance estimates that can be used both as predicted occupancy for moving agents and as adaptive exploration guides for robot control. We evaluate SCOPE with mode-conditioned multimodal diffusion backbones in pedestrian forecasting, crowd navigation, Maze2D control, and real-world Franka Panda manipulation. Across these settings, SCOPE provides fast uncertainty estimation, which leads to better closed-loop performance.

Method
SCOPE turns diffusion trajectories into Gaussian uncertainty tubes for real-time control.
SCOPE extracts aleatoric trajectory uncertainty from diffusion models for two uses:
1) Policy prior: adapt MPPI exploration to the robot's motion flexibility.
2) World model: predict other agents' occupancy for collision-risk evaluation.

Maze2D navigation
A diffusion plan guides MPPI through unseen mazes. SCOPE widens exploration in open regions and narrows it near walls.
MPPI rollout budgets: 4, 8, 16, 32, 64, 128, 256, 512, 1024
Vanilla MPPI
Success rate (%): 43.20, 68.00, 76.75, 82.30, 84.90, 87.35, 89.95, 92.45, 93.75
Mean path length: 16.701, 13.195, 10.846, 9.275, 8.339, 7.505, 6.784, 6.117, 5.573
Log-MPPI
Success rate (%): 32.30, 59.70, 75.30, 82.70, 86.50, 90.00, 92.15, 93.95, 94.55
Mean path length: 20.147, 15.666, 12.509, 10.510, 9.042, 7.884, 7.066, 6.389, 5.994
Diff-MPPI
Success rate (%): 41.10, 64.85, 77.80, 85.25, 88.30, 92.15, 94.45, 95.35, 96.35
Mean path length: 16.518, 12.502, 9.875, 8.290, 7.350, 6.502, 5.981, 5.676, 5.328
SCOPE
Success rate (%): 58.50, 79.50, 87.35, 92.20, 95.35, 97.80, 98.50, 99.15, 99.25
Mean path length: 13.271, 9.978, 8.258, 7.046, 6.123, 5.461, 5.175, 5.060, 4.996

Y-corridor pedestrian navigation
100 episodes per method and density. Success means reaching the goal without collision or timeout.
Pedestrian counts: 3, 6, 9, 12
No future prediction success (%): 80, 72, 70, 65
LKF success (%): 96, 91, 90, 85
MID-UM (no tubes) success (%): 100, 89, 85, 86
MID-MM (no tubes) success (%): 100, 91, 85, 88
MID-Ens-64 success (%): 98, 87, 82, 89
MID-Ens-128 success (%): 99, 90, 82, 86
MID-Ens-256 success (%): 99, 90, 83, 87
SCOPE, no calibration success (%): 100, 94, 86, 88
SCOPE (full) success (%): 100, 96, 93, 92

Real-robot shelf manipulation
A Franka Panda transfers a tomato to the upper shelf, with static obstacles or a moving human hand. Each method runs 30 trials per condition .
Static obstacles, 30 trials per method. Success / collision / timeout counts: Diffuser 14 / 16 / 0. Vanilla MPPI 5 / 24 / 1. Diff-MPPI 15 / 7 / 8. SCOPE 25 / 4 / 1. SCOPE success rate: 83%.
Dynamic interaction, 30 trials per method. Success / any collision / timeout counts: SCOPE without world model 11 / 19 / 0. MID-Ens-64 16 / 14 / 0. MID-Ens-128 14 / 16 / 0. SCOPE 21 / 7 / 2. Environment / human collision counts: 8 / 12, 8 / 8, 9 / 10, 4 / 3. Collision types may overlap. SCOPE success rate: 70%.

Limitations
SCOPE distills uncertainty from a learned diffusion score, so its reliability depends on the backbone. Limited training coverage or poorly separated modes can lead to miscalibrated uncertainty tubes. In real-robot trials, inaccurate human-motion forecasts or yielding into constrained configurations can still cause failures that short-horizon control cannot recover from. Current experiments use low-dimensional state and geometry inputs. Extending SCOPE to images and point clouds remains future work.

Availability
The paper is available at https://arxiv.org/abs/2610.12431. The code link is marked Coming soon.
The figures, videos, and full project description are linked from index.html.