Diffusion-based trajectory optimization has emerged as a powerful planning paradigm, but existing methods require either learned score networks trained on large datasets or analytical dynamics models for score computation. We introduce Behavioral Score Diffusion (BSD), a training-free and model-free trajectory planner that computes the diffusion score function directly from a library of trajectory data via kernel-weighted estimation. At each denoising step, BSD retrieves relevant trajectories using a triple-kernel weighting scheme -- diffusion proximity, state context, and goal relevance -- and computes a Nadaraya-Watson estimate of the denoised trajectory. The diffusion noise schedule naturally controls kernel bandwidths, creating a multi-scale nonparametric regression: broad averaging of global behavioral patterns at high noise, fine-grained local interpolation at low noise. This coarse-to-fine structure handles nonlinear dynamics without linearization or parametric assumptions. Safety is preserved by applying shielded rollout on kernel-estimated state trajectories, identical to existing model-based approaches. We evaluate BSD on four robotic systems of increasing complexity (3D--6D state spaces) in a parking scenario. BSD with fixed bandwidth achieves 98.5% of the model-based baseline's average reward across systems while requiring no dynamics model, using only 1{,}000 pre-collected trajectories. BSD substantially outperforms nearest-neighbor retrieval (18--63% improvement), confirming that the diffusion denoising mechanism is essential for effective data-driven planning.
Figures & tables
Fig. 1: Overview of Behavioral Score Diffusion. Left: A planner initializes from noise; model-free score estimation replaces dynamics rollouts. Center: The BSD denoiser step computes triple-kernel weights (diffusion, context, goal) over the trajectory library, with the noise schedule controlling bandwidth from broad (high noise) to narrow (low noise). Right: Multi-sample shielding reverts violated states, followed by reward-weighted softmax selection. Bottom-right: BSD performance scales gracefully with state dimensionality while nearest-neighbor degrades.
Fig. 2: Main results across four systems of increasing state dimensionality (3D–6D). Each dot shows the mean reward; whiskers indicate bootstrapped 95% confidence intervals (10,000 resamples). Vertical dashed lines mark the MBD (model-based) reference. BSD-fix nearly matches MBD on all systems while substantially outperforming the no-diffusion baseline (NN).
System
Method
Reward [95% CI]
Safety
Time (ms)
Bicycle
MBD
5.95±0.04 [5.94, 5.96]
100%
687
BSD-fix
5.95±0.05 [5.93, 5.96]
100%
2585
BSD
5.63±0.71 [5.42, 5.81]
100%
2876
NN
5.02±0.90 [4.78, 5.27]
100%
< 1
TT2D
MBD
5.73±0.43 [5.60, 5.83]
96%
1181
BSD-fix
5.74±0.41 [5.61, 5.84]
96%
2928
TABLE I: Detailed results: 50 trials per condition. Reward: mean ± std [bootstrapped 95% CI]. Safety: collision- and constraint-free rate.
Fig. 3: Per-trial reward distributions across all four systems (50 trials each). Half-violins show kernel density estimates; individual dots represent single trials; diamonds mark the mean. BSD-fix (red) closely matches MBD (blue) in both location and spread, while NN (grey) exhibits substantially lower and more dispersed rewards. Variance increases with system dimensionality for all methods.
Fig. 4: Performance relative to MBD (%) vs. state dimensionality. BSD-fix (red) maintains near-parity through 5D, while NN (grey) degrades steeply. The widening gap between BSD-fix and NN demonstrates that diffusion denoising becomes more valuable as system complexity increases.
Fig. 5: Paired per-trial reward comparison between MBD and BSD-fix (same random seeds). The diagonal line represents equal performance. High Pearson correlations ( r≥0.70 , up to 0.99 ) indicate BSD-fix tracks MBD faithfully on individual trials, not just in aggregate.
Fig. 6: Safety rate vs. planning time across all systems and conditions. BSD methods achieve comparable safety to MBD at moderate computational overhead. NN has negligible planning time but lowest reward.
Diffusion-based planning generates diverse action sequences or trajectories by progressively denoising candidates conditioned on goals, scenes, and constraints.