CS-Pi: Constraint-Satisfying Flow Policy Sampling

Soumojit Bhattacharya; Oliver Kroemer; Shubham Tulsiani

cs.RONov 5, 2026

CS-Pi: Constraint-Satisfying Flow Policy Sampling

Authors: Soumojit Bhattacharya, Oliver Kroemer, Shubham Tulsiani

Organizations: Robotics Institute, Carnegie Mellon University

Abstract

A flow-matching policy deployed in the field must often satisfy a constraint that was absent from its demonstrations, such as avoiding an obstacle it has never seen. Retraining for every new constraint is impractical, so the constraint must be enforced at inference time. Enforcing it must not discard what the policy learned: an action the policy would rarely sample may avoid the obstacle yet fail the task. We therefore want actions that are both safe, satisfying the constraint, and plausible, remaining likely under the policy. Gradient guidance, the most common approach, steers noisy states, so it must evaluate its cost on the policy's running estimate of the clean action; when the policy is uncertain between passing left or right of an obstacle, this estimate is their average and can pass straight through it. We instead build on rejection and projection. Rejection redraws actions until one is safe: every output is plausible, but the wait is intractable when safe actions are rare. Projection moves an action to the nearest safe one: it is efficient, but the result need not be plausible. Our key insight is that flow-matching sampling lets us combine the two, keeping projection's efficiency and rejection's plausibility. CS-Pi, our draw-project-renoise sampler, projects each draw to the nearest safe action, adds noise back scaled to how far the projection moved it, and re-runs the unmodified policy until a draw is safe. Replacing only the projection step extends CS-Pi to continuous costs. Across a 2D diagnostic, a 7-DoF shelf pick-and-place, an orientation preference, and a zero-shot Franka deployment, CS-Pi reaches 81% safe success on the shelf and 55% on hardware, against 55% and 40% for projection, while staying plausible.

Figures & tables

Figure 1: Sampling a flow-matching policy under a deployment-time constraint. Left: The policy $\pi_{\theta}$ denoises Gaussian noise $\mathbf{a}_{0}\sim\mathcal{N}(\mathbf{0},I)$ ( $t{=}0\to 1$ ) into an action chunk $\mathbf{a}\sim\pi_{\theta}$ , a short sequence of robot motions. Its draws are multimodal: three ways to reach the apple, drawn as paths in the real scene and as modes of the density $\pi_{\theta}(\mathbf{a})$ ; the most likely passes through the obstacle. Middle: Constraints such as obstacle avoidance or velocity and joint limits, and soft costs $L$ , are supplied only at deployment. The obstacle constraint defines a safe set $\mathcal{C}$ , the actions that avoid it. Right: We want actions that are safe ( $\mathbf{a}\in\mathcal{C}$ ) and plausible (likely under $\pi_{\theta}$ ), i.e. draws from $\pi_{\theta}(\mathbf{a})\,\mathbf{1}_{\mathcal{C}}(\mathbf{a})$ , where $\mathbf{1}_{\mathcal{C}}$ is $1$ on $\mathcal{C}$ and $0$ elsewhere: the unsafe mode is removed and the others keep their shape. CS-Pi (ours) targets this distribution without retraining the policy.

Figure 2: Sampling a safe action from a flow-matching policy. Gradient guidance steers each denoising step with a cost gradient evaluated at a proxy and can still violate the constraint. Projection moves the draw to the safe set, yielding an action the policy would rarely sample. Rejection sampling redraws from pure noise until a draw is safe. CS-Pi projects the draw, renoises the projection to time $n=k/(k+\delta)$ set by the projection residual $\delta$ (larger $\delta$ renoises more), and re-runs the policy until a draw is safe and plausible.

Figure 3: 2D Avoidance trajectory distributions, one scenario per method. $20$ rollouts overlaid on the demonstration support (shaded); the obstacles are outlined. Gradient guidance and Trust Sampling visibly leave the support to reach the goal or cut through the obstacle; rejection stays on-support but only a few draws survive; CS-Pi keeps every trajectory in-distribution while routing around the obstacle. The other five scenarios are in App. D .

Method	Safe-SR (%) $\uparrow$	fully-in-supp. $\uparrow$	logL $\uparrow$	NFE $\downarrow$
Unconstrained	$5.2$	$0.800$	$\mathbf{1.66}$	$456$
Gradient guidance	$67.2$	$0.410$	$1.49$	$298$
Trust Sampling	$34.9$	$0.350$	$1.43$	$4421$
Rejection	$79.4$	$0.750$	$\mathbf{1.66}$	$73921$
Projection	$99.8$	$0.640$	$-85.89$	$\mathbf{52}$
SafeDiffuser	$\mathbf{100.0}$	$0.830$	$1.44$	$92$

Table 1: 2D Avoidance , macro over six scenarios (

200

rollouts per cell; best hyperparameter per method). Safe-SR : rollouts reaching the goal collision-free. fully-in-supp. : fraction of rollouts whose every waypoint lies inside the demonstration support. logL : mean log-likelihood under

\pi_{\theta}

Method	Safe-SR (%) $\uparrow$	Goal reaching (%) $\uparrow$	Collision (%) $\downarrow$	logL $\uparrow$	NFE $\downarrow$
Unconstrained	$3.3$	$75.3$	$80.0$	$15.02$	$240$
Gradient guidance ( $\alpha{=}0.20$ )	$32.0$	$32.0$	$\mathbf{0.7}$	$10.42$	$240$
Trust Sampling ( $w{=}0.1$ , $J{=}5$ )	$53.3$	$60.7$	$12.0$	$12.23$	$1104$
Rejection ( $B{=}100$ )	$34.7$	$68.0$	$38.7$	$14.22$	$8745$
Projection	$55.3$	$58.7$	$6.0$	$11.41$	$240$
SafeDiffuser	$13.3$	$16.7$	$19.3$	$7.50^{\dagger}$	$240$

Table 2: Shelf pick-and-place,

5

scenarios,

150

tasks (

30

/scenario). Safe-SR : collision-free rollouts that pick and place the cube. logL : mean log-likelihood under

\pi_{\theta}

(nats/dim; the unconstrained policy scores

15.02

). NFE : flow-model calls per trajectory. † Median: a few diverged SafeDiffuser chunks make its mean uninformative.

Method (best cfg)	Success $\uparrow$	Goal reaching $\uparrow$	logL $\uparrow$	Max EE step (m)	NFE $\downarrow$
Unconstrained	$2.2\%$	$\mathbf{98.9\%}$	$13.46$	$\mathbf{0.07}$	$\mathbf{138}$
Gradient guidance ( $\alpha{=}0.003$ )	$2.2\%$	$70.0\%$	$12.30$	$0.08$	$154$
Gradient guidance ( $\alpha{=}0.03$ )	$6.7\%$	$7.8\%$	$9.62$	$0.40$	$160$
Trust Sampling ( $w{=}0.001$ )	$1.1\%$	$\mathbf{98.9\%}$	$13.44$	$\mathbf{0.07}$	$1379$
CS-Pi + EE-prox ( $k{=}10$ , $K_{\max}{=}3$ )	$\mathbf{65.6\%}$	$\mathbf{98.9\%}$	$10.21$	$\mathbf{0.07}$	$210$

Table 3: 3D-Orient ,

90

rollouts per cell. Success : rollouts with a step simultaneously within

10

cm of the target position and within

0.36

rad of the target orientation. Goal reaching : rollouts ending within

10

cm of the target position. logL : mean log-likelihood under

\pi_{\theta}

; comparisons are meaningful only at comparable success, since the guidance configurations attain only

2.2\%

and

6.7\%

success. Max EE step : largest single-step end-effector displacement. EE-prox : proximal step on the orientation cost (Eq. ( 3 )). NFE : mean number of model evaluations per rollout.

Method	Layout 1	Layout 2	Averaged	Time/step (s) $\downarrow$	logL $\uparrow$
	Safe-SR (%) $\uparrow$
Unconstrained	$20$	$0$	$10$	$0.069$	$-470.2{\pm}1.0$
Gradient guidance	$40$	$10$	$25$	$0.139$	$-464.9{\pm}1.4$
Trust Sampling	$20$	$20$	$20$	$0.135$	$-474.8{\pm}2.1$
Projection	$40$	$40$	$40$	$0.099$	$-473.2{\pm}2.3$
CS-Pi	$\mathbf{60}$	$\mathbf{50}$	$\mathbf{55}$	$0.143$	$-471.8{\pm}1.7$

Table 4: Hardware: pick the apple, place it on the blue plate on a Franka FR3 with

\pi_{0.5}

. Zero-shot;

\mathcal{C}

is obstacle bounding boxes from a ZED 2 RGB-D vision pipeline. Two layouts,

10

trials per method per layout (

20

in total). CS-Pi has the highest safe success rate in each layout and averaged over both. logL is the log-likelihood

\pi_{\theta}

assigns to executed actions (the unconstrained policy’s own draws score

-470.2

; higher is more plausible): CS-Pi stays close to the unconstrained policy, while projection and Trust Sampling fall below it.

Panel of Figure 4 — Figure 4: Hardware rollouts; frames run left-to-right within each strip. Green-bordered strips are CS-Pi , which enforces the deployment-time obstacle constraint and completes the pick-and-place; red-bordered strips are the same $\pi_{0.5}$ policy run unconstrained. Layout 1 (first two): CS-Pi grasps the apple and places it, while the unconstrained policy clips the cup and loses the grasp. Layout 2 (last two): CS-Pi reaches down and lifts the apple cleanly, while the unconstrained policy collides with the obstacle.

Appendix figures & tables27 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 5 — Figure 5: The two Sphere-Avoid layouts (L2 adds six obstacles).

	L1 ( $10$ spheres)			L2 ( $16$ spheres)
Method	Safe-SR	logL	NFE	Safe-SR	logL	NFE
Unconstrained	$8.9$	$2.88$	$10$	$0.8$	$2.84$	$10$
Gradient guidance ( $\alpha{=}0.01$ )	$23.6$	$2.73$	$10$	$4.7$	$2.32$	$10$
Trust Sampling ( $w{=}0.3$ , $J{=}5$ )	$77.0$	$3.03$	$64$	$38.7$	$2.68$	$69$
Rejection	$56.7$	$\mathbf{3.47}$	$388$	$14.1$	$\mathbf{3.19}$	$492$
Projection	$88.2$	$2.67$	$10$	$62.1$	$2.28$	$10$

Appendix

Table 5: 3D Sphere-Avoid .

120

tasks, of which

104

(L1) and

32

(L2) have a safe goal configuration;

8

samples per task. logL : mean log-likelihood under

\pi_{\theta}

(nats/dim; the unconstrained policy scores

2.88

2.84

). † Median: a few diverged SafeDiffuser samples make the mean uninformative (

-119.5

-17.3

). Baselines are at their best swept configuration (App. G ).

Panel of Figure 6 — Figure 6: The five shelf scenarios (blue: cube to grasp, orange: obstacles).

Method	Clutter	Pillar	Flankers	Mid-rung	Low-rung	Mean
Unconstrained	$3.3$	$3.3$	$3.3$	$6.7$	$0.0$	$3.3$
Gradient guidance	$16.7$	$70.0$	$16.7$	$33.3$	$23.3$	$32.0$
Rejection	$20.0$	$83.3$	$20.0$	$33.3$	$16.7$	$34.7$
Projection	$40.0$	$63.3$	$46.7$	$66.7$	$60.0$	$55.3$
SafeDiffuser	$6.7$	$33.3$	$3.3$	$23.3$	$0.0$	$13.3$
SafeFlowMatcher	$43.3$	$46.7$	$53.3$	$56.7$	$23.3$	$44.7$

Appendix

Table 6: Shelf safe success (%) per scenario ,

30

tasks each. CS-Pi ’s mean logL per scenario:

19.29

17.56

17.63

17.08

18.26

. Trust Sampling (

53.3\%

overall) is not broken down by scenario.

Figure 7: 2D Avoidance trajectory distributions, all six scenarios. $20$ rollouts overlaid on the demonstration support (shaded); the obstacles are outlined.

Method	left	right	both	clutt.	narrow	narrower	Mean
Unconstrained	$20.0$	$0.0$	$0.0$	$1.0$	$10.0$	$0.0$	$5.2$
Gradient guidance	$99.5$	$84.0$	$15.0$	$98.0$	$100$	$7.0$	$67.2$
Trust Sampling	$9.0$	$65.5$	$1.0$	$18.0$	$99.0$	$17.0$	$34.9$
Rejection	$60.0$	$95.0$	$70.5$	$71.5$	$86.0$	$93.5$	$79.4$
Projection	$100$	$100$	$100$	$100$	$99.0$	$100$	$99.8$
SafeDiffuser	$100$	$100$	$100$	$100$	$100$	$100$	$100$

Appendix

Table 7: 2D Avoidance safe success (%) per scenario for the baselines ,

200

rollouts each ( CS-Pi :

99.0\%

overall, Table 1 ). Gradient guidance succeeds where one safe corridor dominates ( left , cluttered , narrow ) and fails where the safe set is a narrow passage the policy rarely uses ( both , narrower ).

Method	Setting	Both met (%)	Collision (%)	Mean $\|y-0.7\|$	NFE
Unconstrained	–	$0$	$35.3$	$0.195$	$10$
Gradient guidance	$\alpha{=}0.1$	$0$	$31.3$	$0.194$	$10$
	$\alpha{=}0.3$	$0$	$28.0$	$0.193$	$10$
	$\alpha{=}1$	$0$	$19.3$	$0.189$	$10$
	$\alpha{=}3$	$0$	$0$	$0.178$	$10$
	$\alpha{=}10$	$0$	$0$	$0.150$	$10$

Appendix

Table 8: Constraint and preference on 2D Avoidance ( cluttered ) ,

150

chunks per setting.

Figure 8: Constraint and preference on 2D Avoidance ( cluttered ). No setting of gradient guidance ( $\alpha$ ) or Trust Sampling ( $w$ ) satisfies both the obstacle constraint and the $(y-0.7)^{2}$ preference: low settings collide, high settings stay far from $y_{\mathrm{ref}}{=}0.7$ . CS-Pi meets both.

Figure 9: Saddle-ridge configuration of the orientation cost, rendered on the Franka. Inverse kinematics on a central target $p^{*}=(0.45,0.05,0.50)$ . (A), (B): two minima of $L$ at the same end-effector position ( $0.02$ mm apart) with distinct joint configurations (blue: elbow up; red: elbow rotated and wrist twisted). (C): the configuration $\bar{q}=(q_{1}+q_{2})/2$ that gradient guidance evaluates when its posterior places mass on both branches; the black dot marks its end-effector position, $\mathbf{31}$ cm from the target. The average breaks both the position and the orientation target, although both minima satisfy them.

$k$	$1$	$10$	$20$	$50$	$100$
Safe-SR L1 / L2	$90.1$ / $59.8$	$97.8$ / $77.7$	$99.6$ / $88.3$	$99.4$ / $90.2$	$99.8$ / $94.5$
NFE L1	$72.0$	$24.4$	$20.9$	$19.2$	$18.7$

Appendix

Table 9: CS-Pi on Sphere-Avoid across

k

(

K_{\max}{=}20

, no early stopping; the same evaluation as the no-stopping row of Table 13 ). Table 5 is a separate evaluation, so its values at

k{=}50

differ slightly.

	no post-projection			post-projection
$w$	$J{=}2$	$J{=}5$	$J{=}10$	$J{=}2$	$J{=}5$	$J{=}10$
$0.005$	$2.7$	$6.0$	$4.0$	$29.3$	$30.7$	$32.7$
$0.01$	$3.3$	$4.0$	$11.3$	$30.0$	$33.3$	$34.0$
$0.05$	$9.3$	$38.7$	–	$31.3$	$36.0$	$35.3$
$0.1$	$28.0$	$\mathbf{53.3}$	$51.3$	$33.3$	$32.0$	$43.3$

Appendix

Table 10: Trust Sampling on Shelf: safe success (%) over

w\times J

150

tasks per cell (–: not run). Bold: the reported configuration (logL

12.23

, NFE

1104

Figure 10: Constraint and preference: sweep. Fraction of rollouts meeting both for CS-Pi ( $k$ ), gradient guidance ( $\alpha$ ) and Trust Sampling ( $w$ ); only CS-Pi meets both, for $k\in\{30,100,200\}$ .

Panel of Figure 11 — Figure 11: Trajectories under the constraint and the $(y-0.7)^{2}$ preference , five settings per method. Only CS-Pi ’s sweep contains settings that satisfy both; guidance and Trust Sampling trade one for the other at every setting.

Benchmark ( $k$ )	loops	mean $n$ per iteration	residual first $\to$ last
2D left / right / narrow ( $10$ )	–	$0.64$ – $0.81\to 0.99$ +	$2.7$ – $6.3\to\approx 0.1$
2D both / cluttered ( $10$ )	–	$0.63$ – $0.68\to 0.98$	$6$ – $8\to 0.5$ – $0.7$
2D narrower ( $10$ )	–	$0.73\to 0.80$ (stalls)	$4.1\to 2.9$
2D constraint + preference ( $100$ )	$150$	$0.743,0.881,0.919,0.936,0.946,0.953$	$34.6\to 3.42$
Sphere-Avoid L1 ( $50$ )	$120$	$0.991,0.997,0.999,0.999,0.999,0.999$	$0.478\to 0.0071$
Sphere-Avoid L2 ( $50$ )	$120$	$0.985,0.993,0.995,0.997,0.997,0.997$	$0.767\to 0.0497$

Appendix

Table 11: Time

n

and projection residual per iteration.

Rule	Success	NFE	Iterations
none	$0.825$	$447$	$3.94$
$n>0.8$	$0.864$	$322$	$2.62$
$n>0.9$	$0.863$	$336$	$2.85$
patience	$\mathbf{0.868}$	$\mathbf{206}$	$\mathbf{1.59}$

Appendix

Table 12: Early stopping on 2D Avoidance , means over

11

values of

k

and six scenarios (

K_{\max}{=}20

Rule	$k{=}1$	$k{=}10$	$k{=}20$	$k{=}50$	$k{=}100$
none	$90.1$ / $59.8$ ( $72$ )	$97.8$ / $77.7$ ( $24$ )	$99.6$ / $88.3$ ( $21$ )	$99.4$ / $90.2$ ( $19$ )	$99.8$ / $94.5$ ( $19$ )
$n>0.8$	$90.4$ / $62.9$ ( $54$ )	$88.2$ / $62.1$ ( $10$ )	$88.2$ / $62.1$ ( $10$ )	$88.2$ / $62.1$ ( $10$ )	$88.2$ / $62.1$ ( $10$ )
$n>0.9$	$91.2$ / $64.5$ ( $68$ )	$88.2$ / $60.2$ ( $10$ )	$88.2$ / $62.1$ ( $10$ )	$88.2$ / $62.1$ ( $10$ )	$88.2$ / $62.1$ ( $10$ )
patience	$89.5$ / $64.5$ ( $31$ )	$96.6$ / $75.8$ ( $22$ )	$98.3$ / $81.6$ ( $20$ )	$99.5$ / $84.0$ ( $18$ )	$99.2$ / $88.3$ ( $17$ )

Appendix

Table 13: Early stopping on Sphere-Avoid : safe success L1 / L2 (%) and, in parentheses, L1 NFE.

Rule	$k{=}5$	$k{=}10$	$k{=}20$	$k{=}50$	$k{=}100$
none	$70.7$ ( $398$ )	$78.7$ ( $376$ )	$80.0$ ( $353$ )	$76.7$ ( $306$ )	$69.3$ ( $290$ )
$n>0.8$	$56.0$ ( $255$ )	$56.0$ ( $240$ )	$55.3$ ( $240$ )	$56.7$ ( $240$ )	$55.3$ ( $240$ )
$n>0.9$	$62.7$ ( $272$ )	$62.0$ ( $253$ )	$58.0$ ( $240$ )	$57.3$ ( $240$ )	$57.3$ ( $240$ )
patience	$72.7$ ( $312$ )	$76.0$ ( $307$ )	$80.7$ ( $307$ )	$79.3$ ( $291$ )	$70.7$ ( $280$ )

Appendix

Table 14: Early stopping on Shelf : safe success (%) and, in parentheses, NFE.

Benchmark	Terminal proj.	Safe-SR	logL mean	logL median	Fire rate	NFE
Sphere-Avoid L1	on	$99.9$	$2.87$	$2.81$	$6.7\%$	$20$
	off	$98.1$	$2.87$	$2.81$	–	$20$
Sphere-Avoid L2	on	$91.8$	$2.54$	$2.27$	$46.9\%$	$26$
	off	$69.9$	$2.55$	$2.27$	–	$26$
Shelf	on	$81.3$	$17.96$	$9.68$	$14.7\%$	$271$
	off	$74.0$	$17.59$	$9.94$	–	$270$

Appendix

Table 15: CS-Pi with and without the terminal projection. Fire rate: fraction of sampler calls still outside the tolerance after

K_{\max}

iterations (Shelf: per plan step; Sphere-Avoid : per task, over the tasks with a safe goal configuration).

Benchmark	Fires	Converged	Fired	Projection	Unconstrained
Sphere-Avoid L1	$6.7\%$	$2.81$	$2.90$	$2.41$	$2.74$
Sphere-Avoid L2	$46.9\%$	$2.43$	$2.11$	$2.13$	$2.90$
Shelf	$14.7\%$	$9.36$	$14.70$	$8.45$	$10.00$
Hardware	$4.8\%$	$-469.1$	$-518.0$	$-473.2$	$-470.2$

Appendix

Table 16: Chunks where the terminal projection fires. Fire rate and median logL of converged and fired chunks, with projection and the unconstrained policy as references. Medians, because a few fired chunks dominate the mean.

Method	2D	Sphere-Avoid	Shelf	3D-Orient
Unconstrained	$0.19$	$0.19$	$4.59$	$0.47$
Gradient guidance	$0.52$	$0.58$	$22.1$	$1.56$
Trust Sampling	$8.70$	$2.02$	$89.7$	$12.48$
Projection	$0.08$	$1.99$	$49.0$	–
SafeDiffuser	$0.13$	$7.14$	$164$	–
SafeFlowMatcher	$0.16$	$13.99$	$167$	–

Appendix

Table 17: Wall-clock time per rollout (s) , including flow calls, backward passes, projection and proximal solves.

Benchmark	Reported $k$	Safe-SR (reported $k$ )	Safe-SR ( $k{=}20$ )	Safe-SR ( $k{=}50$ )
2D	$20$	$99.0$	$99.0$	$99.1$
Sphere-Avoid L1	$50$	$99.9$	$99.2$	$99.9$
Sphere-Avoid L2	$50$	$91.8$	$84.4$	$91.8$
Shelf	$50$	$81.3$	$74.7$	$81.3$
3D-Orient	$10$	$65.6$	$64.4$	$64.4$

Appendix

Table 18: Reported versus fixed

k

. Safe success (%) at the

k

reported in Tables 1 , 2 , 3 and 5 , and at a single

k

shared by every benchmark. Over all sweeps, the best

k

is within

1.2

points of

k{=}50

except on L2, where

k{=}100

4.3

points higher (Table 9 ). Shelf varies most between evaluations: four further evaluations at

k{=}50

give

75.8\pm 1.8

(mean

\pm

standard error;

78.7

76.0

78.0

70.7

). Each table is a separate evaluation, so values at the same

k

can differ slightly across tables.

Schedule	Fire rate	Safe-SR	logL median	NFE
adaptive, $k{=}5$ / $10$ / $20$	$15.0$ / $9.5$ / $8.7\%$	$99.2$ / $99.3$ / $99.3$	$1.42$ / $0.38$ / $1.38$	$180$ / $153$ / $145$
constant $n{=}0.5$	$27.7\%$	$99.1$	$0.88$	$181$
constant $n{=}0.8$	$15.6\%$	$99.6$	$-2.23$	$161$
zero noise, $k{=}5$ / $10$ / $20$	$15.0$ / $9.0$ / $8.6\%$	$99.0$ / $99.6$ / $99.8$	$1.16$ / $1.38$ / $1.42$	$183$ / $150$ / $144$

Appendix

Table 19: Schedule for

n

on 2D Avoidance . Fire rate is per plan step. Medians, because a few chunks where the terminal projection fires dominate the mean.

		Acceleration		Jerk
Benchmark	Method	mean	p95	mean	p95
Shelf	Unconstrained	$0.518$	$0.626$	$0.841$	$1.000$
	CS-Pi	$0.541$	$0.651$	$0.882$	$1.073$
	Projection	$0.549$	$0.666$	$0.907$	$1.097$
	Gradient guidance	$0.521$	$0.619$	$0.874$	$1.013$
	Trust Sampling	$0.524$	$0.614$	$0.878$	$0.983$

Appendix

Table 20: Smoothness : per-rollout maximum joint-space acceleration and jerk, mean and 95th percentile over rollouts.

Method	Moved to the surviving mode	median $\Delta\log p$	NFE
Rejection (exact reference)	$92.5\%$	$-0.03$	$284$
CS-Pi	$85.9\%$	$-0.08$	$282$
Projection	$0.0\%$	$-3.56$	$100$
SafeDiffuser ( $\lambda{=}50$ )	$0.0\%$	$-3.56$	$100$
SafeFlowMatcher (published defaults)	$0.0\%$	$-6.42$	$257$

Appendix

Table 21: Blocked-mode diagnostic, blocked subset (

241

draws).

\Delta\log p

: change in exact log-likelihood relative to the unconstrained policy. Every method returns a safe action on all draws.

Figure 12: Blocked-mode diagnostic, left mode blocked. Rejection and CS-Pi move blocked draws to the surviving mode; the other methods stay safe but land in low-likelihood regions of the policy.

Benchmark	Action	CS-Pi	Term. proj.	$\tau$	Rollouts
2D Avoidance	$(v_{x},v_{y})$ , $H{=}16$	$k{=}20$ , $K_{\max}{=}5$	on	$0.01$	$6\times 200$
Sphere-Avoid	B-spline, $28\times 7$	$k{=}50$ , $K_{\max}{=}20$	on	$0.001$	$120\times 8$
Shelf	joint chunk, $16\times 7$	$k{=}50$ , $K_{\max}{=}5$	on	$0.001$	$150$ ( $30\times 5$ )
3D-Orient	7-DoF joints	$k{=}10$ , $K_{\max}{=}3$	off	$0.001$	$90$ ( $30\times 3$ )
Real world	joint velocities	$k{=}5$ , $K_{\max}{=}5$	on	$0.001$	$20$ ( $10\times 2$ )

Appendix

Table 22: Per-benchmark configuration.

\tau

: stopping tolerance. Sphere-Avoid rollouts:

120

tasks

\times\,8

samples. 3D-Orient also uses

\lambda_{ee}{=}3000

and

\beta{=}10^{6}

Method	Trials	Median	p95	p95 / median
Unconstrained	$20$	$116.9$	$210.3$	$1.80$
Gradient guidance	$20$	$90.9$	$322.7$	$3.55$
Trust Sampling	$10$	$178.8$	$333.5$	$1.87$
Projection	$17$	$167.0$	$949.7$	$5.69$
CS-Pi	$20$	$181.6$	$298.7$	$\mathbf{1.64}$

Appendix

Table 23: Hardware smoothness : per-trial maximum end-effector jerk (m/s 3 ), median and 95th percentile across trials.

Constrained Sampling

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Jun 19, 2026J/K move · Enter open · S save

BayesFP: Posterior Estimation for Flow-Based Policies via Feynman-Kac Sampling

Sreevardhan Sirigiri, Weiming Zhi, Fabio Ramos

School of Computer Science, The University of Sydney, Australia · Australian Centre For Robotics, The University of Sydney, Australia · College of Connected Computing, Vanderbilt University, TN, USA.+1

Diffusion Policy Trajectory Generation Flow-Based Generative Modeling Diffusion Posterior Sampling Robot Policy Learning Constrained Sampling+1

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Robots must generate trajectories that remain faithful to learned expert behavior while satisfying safety constraints and task-specific objectives specified only at inference time. We formulate constrained trajectory generation for pretrained diffusion and flow-matching policies as Bayesian posterior sampling, with the learned demonstration distribution as a prior and an inference-time, cost-derived likelihood tilting it toward feasible, optimal trajectories. To sample from this posterior without any retraining of the base policy, we leverage the Feynman--Kac corrector framework, originally formulated for diffusion models, and extend it to deterministic flow-matching policies. The result is a unified, inference-time, retraining-free sampler for diffusion and flow policies. We validate the approach on pretrained Diffusion Policy, GR00T-N1.6, and

π_{0.5}

checkpoints across simulated and real-world manipulation tasks, including planning around non-convex obstacles introduced at inference time, and show improvements over the base

π_{0.5}

on zero-shot tasks.

Abstract

Constrained Sampling

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