Training-Free Diffusion Planning with Analytical Local Scores

Michael Y. Fatemi; Jinhao Liang; Ferdinando Fioretto

cs.ROOct 1, 2026

Training-Free Diffusion Planning with Analytical Local Scores

Authors: Michael Y. Fatemi, Jinhao Liang, Ferdinando Fioretto

Organizations: University of Virginia

Abstract

Path finding and multi-robot motion planning require trajectories that are smooth, goal-directed, and collision-free in environments with complex geometric constraints. Recent diffusion-based planners have shown that trajectory generation can be cast as iterative denoising which has opened the doors to learning-based approaches that can handle multi-modal trajectory distributions and refine entire trajectories. However, a key limitation is that diffusion planners require training on large collections of feasible trajectories, rendering them map-specific, and difficult to deploy when high-quality demonstrations are unavailable. This paper introduces a training-free diffusion-based motion planner that replaces learned global trajectory scores with analytical local scores derived from obstacle, smoothness, velocity, and inter-agent feasibility terms. The proposed idea relies on a key observation: the score of a trajectory can be reconstructed by considering only local interactions between neighboring waypoints and nearby constraints. This structure exploitation yields a decomposed denoising procedure that retains the optimization structure of classical trajectory methods while inheriting the iterative refinement behavior of diffusion models. Experiments on a large collection of complex environments and large multi-agent planning tasks show that the proposed analytical score produces smooth and feasible trajectories within limited computational costs, for example in generating feasible paths for 300+ agents in environments containing 100+ obstacles in under 6 seconds on a GPU, outperforming strong learning-based and optimization baselines, while avoiding the data requirements of learned diffusion planners.

Figures & tables

Figure 1: Training-free diffusion planning. (a) Coarse trajectories connect fixed starts and goals. (b) Analytical local scores combine (i) smoothness, (ii)obstacle avoidance, (iii) inter-agent separation, and (iv) velocity limits. (c) Repeated updates under decreasing noise refine the trajectories without neural training. [Note: the refined paths cross spatially but pass the intersection at different times].

Figure 2: Effective guidance from score candidates

Panel of Figure 3 — Figure 3: Representative 18-robot trajectories generated by TFDP in four benchmark environments.

Map	Robots	TFDP			DGD			MMD			SMD			MPPI
		S	T	P	S	T	P	S	T	P	S	T	P	S	T	P
Basic	6	100	0.18	1.07	100	12.2	1.21	100	27.1	1.12	100	254.3	1.10	100	0.63	1.13
	12	100	0.29	1.08	96	25.0	1.28	100	56.0	1.12	-	-	-	100	0.72	1.16
	18	100	0.45	1.08	96	65.7	1.33	96	86.3	1.13	-	-	-	99	0.81	1.19
Dense	6	100	0.35	1.26	100	12.3	1.26	40	36.5	1.15	100	287.3	1.13	99	0.85	1.19
	12	100	0.66	1.29	100	32.9	1.35	8	62.5	1.15	-	-	-	99	1.09	1.46

Table 1: Success rate (S

\uparrow

, in %), running time (T

\downarrow

, in seconds), and path length (P

\downarrow

). Arrows indicate whether higher (

\uparrow

) or lower (

\downarrow

) values are better. ‘-’ indicates that the method did not complete any instances within the time limit. MPPI times are for GPU ; rest are CPU.

Map	Robots	SDF guidance $(\eta,\alpha)$						TFDP
		$\eta=.1$		$\eta=.2$		$\eta=.5$		$\eta=.5$
		$.05$	$.10$	$.05$	$.10$	$.05$	$.10$	–
Basic	6	100	100	100	100	100	100	100
	12	100	92	100	100	96	88	100
	18	76	80	92	96	92	92	100
Dense	6	92	92	100	96	100	96	100

Table 2: Success rates (%) for SDF guidance and TFDP.

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Map	Robots	Step size $\eta$
		0.01	0.02	0.05	0.10
Basic	6	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.31 $\pm$ 0.06	0.97 $\pm$ 0.02
	12	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.07 $\pm$ 0.03	0.87 $\pm$ 0.04
	18	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.68 $\pm$ 0.06
Dense	6	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.30 $\pm$ 0.06	0.96 $\pm$ 0.02
	12	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.07 $\pm$ 0.03	0.84 $\pm$ 0.04

Appendix

Table 3: TFDP: success probability (

\uparrow

), with Wilson intervals, for smaller step sizes. Bold marks row maxima within this displayed range.

Map	Robots	Step size $\eta$
		0.20	0.50	0.80	1.00
Basic	6	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01
	12	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01
	18	0.97 $\pm$ 0.02	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01
Dense	6	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01
	12	0.96 $\pm$ 0.02	0.98 $\pm$ 0.02	0.99 $\pm$ 0.01	0.98 $\pm$ 0.02

Appendix

Table 4: TFDP: success probability (

\uparrow

), with Wilson intervals, for larger step sizes. Bold marks row maxima within this displayed range.

Map	Robots	Step size $\eta$
		0.01	0.02	0.05	0.10
Basic	6	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.31 $\pm$ 0.06	0.97 $\pm$ 0.02
	12	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.07 $\pm$ 0.03	0.87 $\pm$ 0.04
	18	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.68 $\pm$ 0.06
Dense	6	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.30 $\pm$ 0.06	0.96 $\pm$ 0.02
	12	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.07 $\pm$ 0.03	0.84 $\pm$ 0.04

Appendix

Table 5: SDF guidance with

\alpha=0.1

: success probability (

\uparrow

), with Wilson intervals, for smaller step sizes. Bold marks row maxima within this displayed range.

Map	Robots	Step size $\eta$
		0.20	0.50	0.80	1.00
Basic	6	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01
	12	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01
	18	0.97 $\pm$ 0.02	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01
Dense	6	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01
	12	0.96 $\pm$ 0.02	0.98 $\pm$ 0.02	0.99 $\pm$ 0.01	0.98 $\pm$ 0.02

Appendix

Table 6: SDF guidance with

\alpha=0.1

: success probability (

\uparrow

), with Wilson intervals, for larger step sizes. Bold marks row maxima within this displayed range.

Map	Robots	Step size $\eta$
		0.01	0.02	0.05	0.10
Basic	6	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.31 $\pm$ 0.06	0.97 $\pm$ 0.02
	12	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.07 $\pm$ 0.03	0.87 $\pm$ 0.04
	18	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.68 $\pm$ 0.06
Dense	6	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.30 $\pm$ 0.06	0.96 $\pm$ 0.02
	12	0.01 $\pm$ 0.01	0.01 $\pm$ 0.01	0.07 $\pm$ 0.03	0.84 $\pm$ 0.04

Appendix

Table 7: SDF guidance with

\alpha=0.5

: success probability (

\uparrow

), with Wilson intervals, for smaller step sizes. Bold marks row maxima within this displayed range.

Map	Robots	Step size $\eta$
		0.20	0.50	0.80	1.00
Basic	6	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01
	12	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01
	18	0.97 $\pm$ 0.02	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01
Dense	6	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01
	12	0.96 $\pm$ 0.02	0.98 $\pm$ 0.02	0.99 $\pm$ 0.01	0.98 $\pm$ 0.02

Appendix

Table 8: SDF guidance with

\alpha=0.5

: success probability (

\uparrow

), with Wilson intervals, for larger step sizes. Bold marks row maxima within this displayed range.

Robots	Success $\uparrow$	Time on success $\downarrow$	Mean runtime $\downarrow$
4	0.83 $\pm$ 0.13	2.07 $\pm$ 0.49	2.97 $\pm$ 0.66
6	0.34 $\pm$ 0.17	4.55 $\pm$ 1.91	11.35 $\pm$ 1.42
8	0.14 $\pm$ 0.11	3.37 $\pm$ 2.87	17.30 $\pm$ 1.10
10	0.07 $\pm$ 0.07	–	21.98 $\pm$ 1.09

Appendix

Table 9: RRT-Connect on Dense. Success is a fraction; times are in seconds. Conditional solve time is reported for successful trials when available; “–” denotes an unreported value.

Map	Robots	Linear			Brownian bridge
		S $\uparrow$	T $\downarrow$	P $\downarrow$	S $\uparrow$	T $\downarrow$	P $\downarrow$
Basic	6	98	0.15 $\pm$ 0.00	1.01 $\pm$ 0.00	91	0.14 $\pm$ 0.01	1.02 $\pm$ 0.00
	12	96	0.19 $\pm$ 0.00	1.02 $\pm$ 0.00	85	0.17 $\pm$ 0.01	1.02 $\pm$ 0.00
	18	89	0.25 $\pm$ 0.00	1.02 $\pm$ 0.00	80	0.23 $\pm$ 0.01	1.03 $\pm$ 0.00
Dense	6	92	0.16 $\pm$ 0.00	1.05 $\pm$ 0.00	86	0.15 $\pm$ 0.01	1.07 $\pm$ 0.00
	12	87	0.23 $\pm$ 0.00	1.06 $\pm$ 0.00	72	0.21 $\pm$ 0.01	1.07 $\pm$ 0.00

Appendix

Table 10: TFDP with linear and Brownian-bridge initialization. Bold compares the two initializers within each row and metric; path length is ranked only for initializers with nonzero success.

Map	Robots	Voronoi			RRT
		S $\uparrow$	T $\downarrow$	P $\downarrow$	S $\uparrow$	T $\downarrow$	P $\downarrow$
Basic	6	100	0.30 $\pm$ 0.00	1.20 $\pm$ 0.00	100	0.40 $\pm$ 0.02	1.09 $\pm$ 0.01
	12	100	0.47 $\pm$ 0.00	1.22 $\pm$ 0.00	99	0.54 $\pm$ 0.02	1.09 $\pm$ 0.00
	18	99	0.71 $\pm$ 0.00	1.24 $\pm$ 0.00	100	0.82 $\pm$ 0.04	1.12 $\pm$ 0.00
Dense	6	96	0.39 $\pm$ 0.00	1.29 $\pm$ 0.01	99	0.67 $\pm$ 0.02	1.32 $\pm$ 0.02
	12	90	0.63 $\pm$ 0.00	1.32 $\pm$ 0.00	98	1.13 $\pm$ 0.02	1.37 $\pm$ 0.01

Appendix

Table 11: TFDP with Voronoi and RRT initialization. Bold compares the two initializers within each row and metric; path length is ranked only for initializers with nonzero success.

Map	Robots	Linear			Brownian bridge
		S $\uparrow$	T $\downarrow$	P $\downarrow$	S $\uparrow$	T $\downarrow$	P $\downarrow$
Basic	6	98	0.65 $\pm$ 0.01	1.03 $\pm$ 0.00	99	0.60 $\pm$ 0.03	1.04 $\pm$ 0.00
	12	97	0.65 $\pm$ 0.01	1.04 $\pm$ 0.00	95	0.61 $\pm$ 0.03	1.04 $\pm$ 0.00
	18	96	0.68 $\pm$ 0.01	1.05 $\pm$ 0.00	87	0.61 $\pm$ 0.03	1.05 $\pm$ 0.00
Dense	6	92	0.66 $\pm$ 0.01	1.08 $\pm$ 0.01	83	0.62 $\pm$ 0.03	1.10 $\pm$ 0.00
	12	78	0.67 $\pm$ 0.02	1.12 $\pm$ 0.00	63	0.62 $\pm$ 0.04	1.14 $\pm$ 0.00

Appendix

Table 12: MPPI with linear and Brownian-bridge initialization. Bold compares the two initializers within each row and metric; path length is ranked only for initializers with nonzero success.

Map	Robots	Voronoi			RRT
		S $\uparrow$	T $\downarrow$	P $\downarrow$	S $\uparrow$	T $\downarrow$	P $\downarrow$
Basic	6	100	0.69 $\pm$ 0.03	1.21 $\pm$ 0.01	100	0.63 $\pm$ 0.01	1.13 $\pm$ 0.01
	12	100	0.80 $\pm$ 0.03	1.26 $\pm$ 0.00	100	0.72 $\pm$ 0.01	1.16 $\pm$ 0.01
	18	99	0.93 $\pm$ 0.03	1.29 $\pm$ 0.00	99	0.81 $\pm$ 0.02	1.19 $\pm$ 0.00
Dense	6	95	0.79 $\pm$ 0.03	1.34 $\pm$ 0.01	99	0.85 $\pm$ 0.02	1.46 $\pm$ 0.02
	12	89	0.95 $\pm$ 0.03	1.40 $\pm$ 0.00	99	1.09 $\pm$ 0.02	1.53 $\pm$ 0.01

Appendix

Table 13: MPPI with Voronoi and RRT initialization. Bold compares the two initializers within each row and metric; path length is ranked only for initializers with nonzero success.

Map	Robots	MPPI time (s) $\downarrow$
Basic	6	8.18 $\pm$ 0.13
	12	19.43 $\pm$ 0.26
	18	32.36 $\pm$ 0.47
Dense	6	10.84 $\pm$ 0.26
	12	24.85 $\pm$ 0.54
	18	40.98 $\pm$ 0.86

Appendix

Table 14: MPPI runtime on the same CPU as TFDP; lower is better.

Scenario	TFDP, noisy $\uparrow$	TFDP, noiseless $\uparrow$
Dense (6 agents)	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01
Dense (12 agents)	0.98 $\pm$ 0.02	0.98 $\pm$ 0.01
Dense (18 agents)	0.93 $\pm$ 0.03	0.95 $\pm$ 0.02
Basic (6 agents)	0.99 $\pm$ 0.01	1.00 $\pm$ 0.00
Basic (12 agents)	0.99 $\pm$ 0.01	0.99 $\pm$ 0.01
Basic (18 agents)	0.99 $\pm$ 0.01	1.00 $\pm$ 0.00

Appendix

Table 15: Success fractions with noisy and deterministic annealed refinement. Bold marks the higher central value in each row, including ties.

$\sigma\backslash T$	0.0001	0.0005	0.001	0.005	0.01	0.05	0.1	0.3	0.5	0.8
0.01	0.28	0.28	0.24	0.28	0.24	0.28	0.28	0.28	0.28	0.40
0.025	0.32	0.32	0.36	0.36	0.36	0.32	0.36	0.36	0.56	0.56
0.05	0.40	0.40	0.40	0.36	0.40	0.36	0.40	0.64	0.68	0.72
0.075	0.48	0.48	0.48	0.48	0.48	0.44	0.44	0.60	0.68	0.60
0.1	0.44	0.44	0.44	0.44	0.44	0.44	0.44	0.72	0.64	0.40
0.125	0.48	0.48	0.48	0.48	0.48	0.52	0.60	0.60	0.64	0.32

Appendix

Table 16: TFDP success fraction (

\uparrow

) on Dense. Rows vary initial

\sigma

and columns vary initial

T

; bold marks the best temperature in each row, including ties.

$\sigma\backslash T$	0.0001	0.0005	0.001	0.005	0.01	0.05	0.1	0.3	0.5	0.8
0.01	0.72	0.72	0.72	0.72	0.72	0.72	0.72	0.72	0.76	0.68
0.025	0.72	0.72	0.72	0.72	0.72	0.72	0.72	0.76	0.72	0.88
0.05	0.72	0.72	0.72	0.72	0.72	0.72	0.72	0.80	0.88	0.96
0.075	0.72	0.72	0.72	0.72	0.72	0.76	0.76	0.84	0.92	0.88
0.1	0.72	0.72	0.72	0.72	0.72	0.72	0.76	0.88	0.96	0.76
0.125	0.76	0.76	0.76	0.76	0.76	0.72	0.76	0.92	0.88	0.48

Appendix

Table 17: TFDP success fraction (

\uparrow

) on Basic. Rows vary initial

\sigma

and columns vary initial

T

; bold marks the best temperature in each row, including ties.

Figure 4: Stress-test layouts: Bottleneck (left) and Standoff (right, labeled “Deadlock”). Both have an opening wide enough for one robot.

Robots	MPPI success $\uparrow$	TFDP success $\uparrow$
2	0.96 $\pm$ 0.02	0.98 $\pm$ 0.02
4	0.76 $\pm$ 0.05	0.95 $\pm$ 0.03
6	0.51 $\pm$ 0.06	0.89 $\pm$ 0.04
8	0.20 $\pm$ 0.05	0.69 $\pm$ 0.06
12	0.02 $\pm$ 0.02	0.31 $\pm$ 0.06

Appendix

Table 18: Bottleneck success fractions. Bold marks the higher central value in each row.

Robots	MPPI success $\uparrow$	TFDP success $\uparrow$
2	0.96 $\pm$ 0.02	0.99 $\pm$ 0.01
4	0.78 $\pm$ 0.05	0.92 $\pm$ 0.03
6	0.51 $\pm$ 0.06	0.85 $\pm$ 0.05
8	0.23 $\pm$ 0.05	0.70 $\pm$ 0.06
12	0.03 $\pm$ 0.02	0.32 $\pm$ 0.06

Appendix

Table 19: Standoff success fractions. Bold marks the higher central value in each row.

Window	$\sigma=0.025$	$\sigma=0.05$	$\sigma=0.1$	$\sigma=0.2$
1	0.8391	0.8679	0.9317	0.9036
3	0.4587	0.3849	0.3572	0.3734
5	0.4169	0.3118	0.2566	0.2712
7	0.4752	0.2988	0.2299	0.2334
9	0.4464	0.3006	0.2184	0.2037

Appendix

Table 20: Final-epoch mean squared error (

\downarrow

) of the diagnostic score estimator. Bold marks the lowest error across window sizes for each noise level.

Map	Robots	S $\uparrow$ (%, 95% CI)	T $\downarrow$ (s)	P $\downarrow$
Basic	6	$100\;[\,98.5,\,100.0\,]$	$0.18\pm 0.02$	$1.07\pm 0.05$
	12	$100\;[\,98.5,\,100.0\,]$	$0.29\pm 0.08$	$1.08\pm 0.04$
	18	$100\;[\,98.5,\,100.0\,]$	$0.45\pm 0.07$	$1.08\pm 0.03$
Dense	6	$100\;[\,98.5,\,100.0\,]$	$0.35\pm 0.09$	$1.26\pm 0.15$
	12	$\phantom{0}99\;[\,96.8,\,99.7\,]$	$0.66\pm 0.07$	$1.29\pm 0.09$
	18	$\phantom{0}96\;[\,92.7,\,97.9\,]$	$1.06\pm 0.19$	$1.32\pm 0.03$

Appendix

Table 21: TFDP success with

95\%

Wilson intervals, runtime, and path length. Runtime and path length use mean

\pm 2

standard errors; gray identifies TFDP results.

Panel of Figure 5 — Figure 5: Example benchmarks used for our experiments.

Figure 6: Generated trajectories on the larger map used for the scalability experiment. Colored curves show robot paths; red regions show obstacles.

Figure 7: Fraction of feasible plans over refinement iterations on Dense (top left), Basic (top right), Room (bottom left), and Shelf (bottom right). Blue denotes TFDP; orange and green denote CHOMP-style SDF guidance at step sizes $0.5$ and $0.2$ , respectively.

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