SafeFlowMatcher: Safe and Fast Planning using Flow Matching with Control Barrier Functions
Authors: Jeongyong Yang
Seunghwan Jang
SooJean Han
Venue: ICLR 2026 Poster (2026)
Keywords: Flow matching
Control Barrier Functions
Planning
Control
Primary area: applications to robotics, autonomy, planning
TL;DR: We propose SafeFlowMatcher, a novel method for safe and fast planning that couples flow matching with control barrier functions via a two-phase prediction–correction integrator

Abstract: Generative planners based on flow matching (FM) produce high-quality paths in a single or a few ODE steps, but their sampling dynamics offer no formal safety guarantees and can yield incomplete paths near constraints. We present SafeFlowMatcher, a planning framework that couples FM with control barrier functions (CBFs) to achieve both real-time efficiency and certified safety. SafeFlowMatcher uses a two-phase prediction-correction (PC) integrator: (i) a prediction phase integrates the learned FM once (or a few steps) to obtain a candidate path without intervention; (ii) a correction phase refines this path with a vanishing time‑scaled vector field and a CBF-based quadratic program that minimally perturbs the vector field. We prove a barrier certificate for the resulting flow system, establishing forward invariance of a robust safe set and finite-time convergence to the safe set. In addition, by enforcing safety only on the executed path—rather than all intermediate latent paths—SafeFlowMatcher avoids distributional drift and mitigates local trap problems. Moreover, SafeFlowMatcher attains faster, smoother, and safer paths than diffusion- and FM-based baselines on maze navigation, locomotion, and robot manipulation tasks. Extensive ablations corroborate the contributions of the PC integrator and the barrier certificate.

Published as a conference paper at ICLR 2026
SAFEFLOWMATCHER: SAFE AND FAST PLANNING USING FLOW MATCHING WITH CONTROL BARRIER FUNCTIONS
Jeongyong Yang', Seunghwan Jang" SooJean Han
Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea faeiryu2238, 38, jsh991124, soojean) j (eaaekkaaitt.......... ac.kr "Equal contribution. *Corresponding author.
ABSTRACT
Generative planners based on flow matching (FM) produce high-quality paths in a single or few ODE steps, but their sampling dynamics offer no formal safety guarantees and can yield incomplete paths near constraints. We present SafeFlow- Matcher, a planning framework that couples FM with control barrier functions (CBFs) achieve both real-time efficiency and certified safety. SafeFlowMatcher uses a two-phase predictionccorecctin
(PC) integrator: (i) prediction phase in- tegrates the learned FM once (or a few steps) to obtain candidate path without intervention; (ii) a correction phase refines this path with vanishing time-scaled vector field and a CBF-based quadratic program that minimally perturbs the vector field. We prove barrier certificate for the resulting flow system, establishing for- ward invariance of robust safe set and finite-time convergence to the safe set. addition, by enforcing safety only the executed path nath-rathhr than all interme- diate latent cal trap problems. Moreover, SafcFlow.Mtchhe attains faster, smoother, and safer paths than diffusion- and FM-based baselines on maze navigation, locomotion, and robot manipulation tasks. Extensive ablations corroborate the contributions of the PC integrator and the barrier certificate. Code is available at the project page
avoids distributional drift and mitigates lo-
INTRODUCTION
Robotic path planning must simultaneously achieve real-time responsiveness and strong safety guar- antees. Recently, generative models such diffu- sion (Ho et al. 2020; Dhariwal Nichol, [02T]] 202 lb) and flow matching (FM) (Lip man et al. 2023) have gained attention for path planning. thanks to their expressive modeling of multi-modal action distributions (Carvalho et al.)
Trapped
matee Jonngidtion
Intogaatinn
Song et al.
Other certifiatice hasd Mirlodds
Prediction Intggaitoo
(cwn)
0.1
Correction
2023 Braun al. 2024) and low-latency infer- ence Qureshi et al. 2019 Liu et al. (022)) com-
pared classical sampling- and optimization-baser planners. However, the sampling dynamics of these models are governed by implicitly learned rules and can produce paths that violate physical safety con- straints, leading to task interruptions or collisions. Therefore, integrating certified safety into genera- tive planning is essential for deployment in real- world robotic systems.
Figure 1: Overview of SafeFlowMMathher Versus Existing Certification-Bass Meth- ods. Directly constraining intermediate sam- ples during generation (top) can cause paths to distorted or trapped, whereas Safe- FlowMatcher (bottom) decouples generation and certification, producing complete and certified-safe path.
Several approaches have attempted to enforce safety in generative planning. Safety-guidance methods regulate the sampling process through learned safety scores (often called guidance, e.g., classifier--free Dhhrriae & Nichol 2021 Ho & Salimans 2021), value/reward guid-

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Published as a conference paper at ICLR 2026
ance YYang et al., 2024 Chen al. 2024)), but their reliance data-driven proxy prevents them from providing strong safety guarantees. More explicitly, certification
methods incorporate functions such as Control Barrier Functions (CBFs) directly into the generative process (Xiao al.) 2025). Unlike sondance-based approaches, these methods can guarantee safety at deployment with- requiring additional training. However, key challenge in such certification- -aased methods is a semantic misaligmment: certification concerns the executed physical path (its waypoints over the horizon), whereas interventions are often applied intermediate latent states that are never executed. Constraining such latents is unnecessary for certification. As result, repeated interventions distort the learned flow and often yield incomplete (locally trapped) paths. Finally, although diffusion sam- plers can be accelerated (Lu et al.) 2022, Zhang & Chen, 2022 Liu et al. (022). their SDE-based denoising requires many steps, making real-time planning expensive. In contrast, FM casts sampling as deterministic ODE integration, generating accurate paths in a single or few steps.
To address these limitations, we propose flw.aacer,,,,,r a planning framework that combines flow matching with CBFs, particularly for finite-time convergence CBFs, to achieve certified safety before the completion of generation, while maintaining the efficiency of FM. Our key idea is a prediction-correctton (PC) integrator that decouples distributional drift from safety certification. In the prediction phase, we propagate the flow once (or a few steps) to obtain a candidate path without any safety intervention. In the correction phase, we refine this path by (i) compensating for integra- tion error through a modified vector field, and (ii) enforcing safety through CBFs. Rather than con- straining all intermediate samples from pure noise to the target during prediction, SafeFlow enforces safety only in the correction phase. This preserves the native FM dynamics and prevents distributional drift when generating the target path. Also, it avoids local traps caused by repeatedly pushing intermediate waypoints onto the barrier boundary and stalling near safety constraints. In summary, our main contributions are as follows:
We introduce SafcFlow.Matcher. novel planning framework that integrates finite-time con- vergence CBF-based certification with flow matching to enforce hard safety constraints, while preserving the efficiency of flow matching.
We propose prediction-correctin fication: FM first generates paths without intervention, and then CBF-based corrections enforce finite-tima convergence to the safe set while compensating for integration errors.
integrator that decouples path generation from certi-
We validate Satchlow in maze navigation, locomotion, and robot manipulation tasks with extensive ablation studies, showing consistent improvements over both FM- and diffusion-basett planners in efficiency, safety, and path quality.
2 RELATED WORK & PRELIMINARIES
2.1 FLOWMATCHER: FLOW MATCHING FOR PLANNING
FM has recently been proposed as a powerful alternative to diffusion, originally in the image gen- eration domain (Lipman et al.) 2023 |Song et al. (2021b), and has shown promise for efficient path planning and robotic control Ye & Gombolay] 2024] Zhang & Gienger| 2024 Chisari et al. 2024 Xing et al. 2025). Unlike diftusion, FM directly learns time-varying vector field that maps noise to the target distribution via forward integration, making the sampling process efficient and flexible.
We adapt standard flow matching (FM) (Lipman et al. 2023) to the planning context. Let H E N be the planning horizon and H {O..... H). A path is a stacked vector T (T", T HH) DH+1 C [ where each waypoint rK E 'D C Rd encodes the state at step k.
Let v,(::)= DH+1 - DH+1 be a time-dependent vector field. The flow sb : [0, DH+1
DH++1 is defined as the solution of the ODE
d di (x(()) (1))
Vo(T) =T
which transports a simple prior po (e.g. NNO, 7)) to a target pt. Following conditional flow matching (CFM), we train v)(=;)) by regressing it to conditional vector field that generates a fixed conditional probability path. We adopt the optimal transport (OT) path Pe(T Ti) = N ([T: Pe(T)), T),,, (e(T)) =t,, oz -t, whose generating OTT-coniitional vector field is
ue(T I 7:) Ti -7
1-1
2

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Let denote the data distribution over the target paths T1- Sampling ~~ Unif 1]. To ~op T1~ and defining 7: A (To) (1 T & 7i (conditioned TI), have by 2 that ( (1)
Ti To- Hence, we train (=(:;)) with the CFM loss:
( E, o( Po(T)) (e(1) (Ti To) 1
(3)
Further details are in Lipman et al. (2023).
For numerical integration, we discretize 0=ty < ... < tr = 1 with the sampling horizon T N (Collectively T(T) = (1o.-..,(T) (Tr) and define step sizes Al; 1++1 4. We define T-step in-
tegrator (1 DH++ + DHI+1 (e.g.. Euler integrator |
which integrates the flow matching
dynamics from To to 7i as
T-1
(T) 0-s1
(To) = Tu Σ
At, 04 (Tr,:).
(4)
2.2 CONTROL BARRIER FUNCTIONS
Safety filters (Hsu et al. 2023 Wabersich et al. 2023) are a real-time intervention mechanism to ensure that an autonomous agent operates within some predefined safe sets, overriding its nominal behavior only when it is about to violate the sets. Various approaches exist for constructing safety filters, but among these, control barrier functions (CBFs) (Ames et al., 2019) are especially popular, as they provide systematic way to guarantee forward invariance of safe sets by solving a real-time optimization problem at each control step. Additional recent works on CBFs, including non-convex safe sets and learning-based CBFs, are summarized in Appendix A
Here, we review only the standard finite-time convergence CBF preliminaries that are necessary for the rest of this paper. To this end, we consider an arbitrary control-affine system
X
f(x+) + g(x)
(5)
where x, D C Rdd u, E 2 C Rd, and f Rd - Rd and g Rd Rddd are locally Lipschitz continuous.
Define the safe set C as the superlevel set of continuouslly-dfffeenntaall (C*) function b b: :D TR,
[x+ E D b(x,) 2 0}.
(6)
System safety is often mathematically prescribed by ensuring that a system's 'ss state safely converges to the targeted safe set within finite time.
Definition (Finite-Time Convergence CBF) Given the system 5 and the safe set O C func-
tion b is called finite-time convergence CBF if there exist parameters p E [0, 1) and > 0 such that for all X, E D.
sup [L)b(x;) + Lgb(x;) Ju + + sgn(b(x))))b(x) ) 0,
u.Eu
where b,,()x)) (b(x,) f(x,) and b(x,) Vb(x,) T (xx,) denote the Lie derivatives of along and g, respectively.
Lemma 1 (Forward Invariance of the Safe Set) Define CBF b as in Definition 7 such that the initial state satisfies b(x,) 2 0. Any Lipschitz continuous controller u, that satisfies condition ?
ensures forward invariance of the safe set C, i.e. b(x.) 2 0 for all t 0.
Lemma D ensures that once the state first enters the safe set, it remains there thereafter. To select
control input that guarantees forward invariance of C while remaining close possible reference control input
- arg min u.ell
| 1 subject to Lyb(x,) + L,b(x;)u, + e- sgnss(((()) 2 0. (8)
This means the optimal solution o. is the minimally modified control that guarantees the forward invariance of the safe set C. Moreover, based on the finite-time stability theorem (Bhat & Bernstein, 2000), the finite-time convergence CBF can be used to ensure that states not only remain within the safe set but also reach within finite-tinn time (Li et al. 2018 Srinivasan et al. 2018).
A Itermatively, higher-order ODE solvers he used.
3

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3 SAFEFLOWMATCHER
Here, we present SafeFlowMann fowMattrw,,,,,r, safe and fast planning framework that couples flow matching with certified safety settings where neither the dynamics nor cost map are known. First, Sec- (PC) integrator which decouples genera-
tion 3.1 we introduce two-phase prediction-correcll tion and certification. Next, in Section 3.2 we formalize safety for MatchlowMatcher by employing control barrier functions (CBFs) and derive conditions that guarantee forward invariance and finite- time convergence to the safe set. The pscudocode of SafeFlow.Matche is in Algorithm [ and full generation processes for two Maze environments visualized in Appendix F5
We introduce a crucial problem in non-autoregrressvve planners, par- ticularly for generative- based planner. As shown in Figure non- autoregressive planners may fail to generate a complete path after planning when using CBFs. Although the resulting path remains safe (does not exceed safety constraints), it may be unable to reach the goal because certain waypoints become locally trapped near the barrier boundaries and cannot escape within the finite sampling or integration time. We will show that SafeFlow RaffehooMMtchher Matcher can effec- tively resolve this issue using PC integrator.
Definition 2 (Local Trap) A local trap problem occurs during the planning process if there exists E H such that || - - - Ç. where ( >> 0 is user-defined threshold depending on the planning environment.
Figure 2: Local trap. Exam- ple of a local trap in maze en- vironment.
Algorithm Matcher
Input: learned velocity field vz(:; (); 0), prediction and correction horizon 7 , planning horizon H. CBF parameters (e, p), robustness parameter 6, and scale constant a
Output:
Safe path Tí
Phase 1: Prediction
1:
Sample initial noise TO ~ 1)
2: Compute predicted path 7f - ( by
Phase 2: Correction
3: Initialize corrected path TO - TP
4: for each correction step t T do
5: Ür < a (1) (T 0)
6: for each waypoint & E H do
7:
Solve QP
(17) to obtain (u ;+k", using 0z
8: end for
9: Update velocity time-scaled flow dynamics 13 with ot =
{u 0.* ..... Up H
end for
11: Return safe final path 7f
3.1 EREDICTION-CORRECIO INTEGRATOR
SaftFlow divides the integration process into two phases: a prediction phase that generates an approximate path without considering safety, and correction phase that refines the path by reducing integration error and adding safety constraints. Let T! D C
dd for E E {p. c} denote the paths in the prediction and correction phases, with waypoints T E.k E DCC Rd for k E H. Additionally, we denote by = + the total sampling horizon, where T and T are the number of sampling (integration) steps allocated to the prediction and correction phases, respectively.
The Prediction phase aims to quickly approximate the target path starting from pure noise T ~ (0, /), without considering safety constraints. Starting from the noise, we Fuler integration
Thhe definition is slightly different from that of SafeDiffuser Xiao et al. 2025 to capture broader class of failure cases. See Appendix for the details.
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obtain the solution of the flow matching dynamics 1
F (7)) +E,
where of is the exact solution of the flow matching dynamics and is the Euler integration (pre- diction) error. To balance computational efficiency and reliability, we select small 7 (typically TP 1) that places 7i sufficiently close Tí. making it suitable initialization for the correction phase.
The Correction phase starts from the path in the prediction phase Tp Tf. unlike To in the prediction phase. In this phase, the path is refined by (i) reducing the discretization error e and (ii) enforcing safety constraints.
To achieve (i), we introduce the vanishing time-scaled flow dynamics (VTFD)
drp
di a (1- Ce(T 0) (T(T{;0),
(10)
where the factor (1-t) gradually suppresses the vector field t++1 with scaling constant 0. Intuitively, this produces contraction effect: the path is driven toward the target direction in the early correction steps. while the dynamics become increasingly stable near t 1, preventing drift and allowing the prediction error to decay. This mechanism is formalized in lemmapand LandLenm Lemma
Lemma Assume the prediction error & ~ Pr. has a symmetric, zero-mean distribution (e.g-, Gaus- siun) and that, in neighborhood of 0, the negative log-density - -og Pe is C² with positive- definite Hessian A > (i.e., locally strongly convex). In addition, assume the target log-density log Pı is C Suppose the correction phase is initialized near the target Ti:
Ti . + (1-11) e)
0
(11)
Then, E T = -I + 00
We empirically verify the validity of the symmetric zero-mean assumption on the prediction error in Appendix B.I (TT) natural result under optimal transport, since OT path approaches Ti as t + 1. Lemma ensures that the posterior expectation contracts toward the target.
Lemma 3 Under the assumptions of Lemina
let e, - Tf Tí. If the flow dynamics follow the vanishing time-scaled flow dynamics (10). then as t + 1,
2
e, = 0(1 1)9) + (e 0(1) )
(12)
Lemma 3 implies that VTFD reduces the prediction error of 7.. See the proofs of Lemma and Lemma Appendix B.2
3.2 CONTROL BARRIER CERTIFICATE FOR SAFEFLOW
To ensure the safety constraints hold during the correction phase, we introduce an additional pertur- bation minimally intervene the flow dynamics (T)::
drf
8g (T{ O) + Au,
(13)
dt
where i is VTFD defined in 10 and Au, (Auf, Au! Auf") E Rdx((H+1) (Auf Rd)) is perturbation term that enforces safety constraints. Importantly, the safety constraint is applied in waypoint-wise fashion: the CBF condition is enforced independently for each waypoint T,ch that remains within safe set C. Thus, we can split the dynamics (13 into
drc
dt o (Tf: 0) + Au u
(14)
where o (Tf: 0) denotes the k-th column of 6, (Tf;0) For notational simplicity, we denote the right- hand side by u, = fuf u.,... u ddx(H+1) (u Rd). 3 We now formalize the concept of safety in flow matching using finite-time flow invariance.
14 is control-afiine system with drift f(r(**) = P and input matrix = I. Thus, at the waypoint level, the structure coincides with the standard control-affine system used in Section 2.2
5

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Definition 3 (Finite-Time Flow Invariance) Let b: + be C' function. The system 13 is
finite-time flow invariant if there exists iy [0, 1] such that b( b(T(-²) 2 0 for all ho E H. Vt ig-
Theorem (Forward Invariance for SafeFlowMatcher) Let b: D - R be a Ca function, and de-
fine the robust safe set Cg & {Tec E ((TC.)= 2 8) for some 0. Suppose the system 13) is
controlled by uy satisfying the following barrier certificate for 0 < p < 1, < 0:
Vb(TT..)) Tuk + (e-sgn(b(TT 50)6(6))((( 81p w w 2 0, Ak H. [0.
(15)
Here, w [0, - Rzz is a monotonically decreasing function with w 0 for all E [he 1]
(tw E [0, 1)), and r 0 is slack variable. Then the flow matching 13 achieves finite-time flow invariance on Cg.
The weights w serve as functions that relax the CBF constraint in the early refining phase, providing numerical stability by preventing infeasibility and reducing abrupt changes in the QP solution. Since w vanishes for & 2 tw. the relaxation term has no effect afterwards, ensuring that the final path satisfies certified safety.
Proposition 1 (Finite Convergence Time for SafeFlowMatcher) Suppose Theorem hholds. Then for any initial waypoint Tt E \ Cos the waypoint T, converges the safe set Cs within finite time
(6 b( Tiw 1TP
T tw +
(16)
c(1 p)
and remains in the set thereafter.
Proposition alllows us to select parameters and to guarantee flow invariance on the robust safe set Cs before the time (I6p. The proofs of Theorem [n and Proposition I are Appendix C
In order to enforce the invariance of the safe set C5 with minimum intervention during planning, we
solve quadratie program (QP) analogous to (8) each sampling time t and planning step k:
fc u Pt
arg min |!u|
o (TT::0]] + rt subject
(15).
(17)
where ur and 8 (T;;0) are defined in 14 Since the QP 17 equivalent to a Fuclidean projec- tion problem with linear inequalities, closed-form solutions are available when it has at most two inequalities (Luenberger) [997; Boyd & Vandenberghe 2004). Moreover, the computational time can be reduced further by decreasing the correction horizon T or balancing (TP, T'), as discussed in Appendix F.
Remark The PC integrator brings To closer to the barrier boundary after the prediction phase. By Proposition 7 this improved initialization reduces the required convergence time, allowing us a wider range of choices for (p, <), and more stable control inputs. We empirically validate this in Appendix
F2
Remark The relaxation term mainly necessary in environments where the planner is prone to becoming locally stuck due to complex safety constraints. In particular, it is essential in Maze2D, where the safe set highly non-convex, leading to frequent local traps. In contrast, in relatively open convex environments such as locomotion robot manipulation tasks in our experiments, the relaxation is typically unnecessary. In such cases, the relaxation term w remains zero, and the slack variable r can be removed from 15).
4 EXPERIMENTS
We evaluate SafeFlow through experiments designed to answer three key questions:
Does SafchlowMMtcher outperform state-of-the-arr generative model based safe planning baselines in terms of safety, planning performance, and efficiency?
2. Does SafeFlowMatchee really require a two-phase (prediction and correction) approach?

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3. How well can SafeFlowMMtcher generalize to complex and high-dimensional tasks (e.g., robot locomotion and manipulation)?
We conduct experiments
variety of planning domains: ii) Maze navigation
(maze-large-vil) (ii OpenAl Gym locomotion (Walker2D-Med) um um Eeppee-x------eee -2,
(Brockman al. 2016: Todorov al. 2012). and (iii) robot
manipulation task (block stacking) (Janner et al. 2022
To fairly evaluate our proposed method, extend SafeDiffuser (Xiao et al.
2025 beyond its orig- inal DDPM sampler. We introduce three additional safety-aware variants. For the first and second variants, adapt DDIM (Song al., 2021a) into versions, SafeDDDM()=000 & 1.0), which share the same weights as SafeDiffuser; here, controls the level of sampling randomness. The last variant we develop is SafeFM, a flow-macching counterpart SafeDiffuser which uses the same weights as Saathherr but enforces safety directly during sampling and without the Ieffiction-ooreecti integrator. When safety constraints are disabled, we drop the "Safe" prefix. Additional details on experimental settings are provided in Appendix E.
For safety, we report Burrier Safety (BS) per constraint, the minimum value the barrier function (which should remain non-negative), and Trap Rate, the rate of local trap occurrences. For planning quality, we measure the overall Score, the average path Curvature (s), and the average path Accel- eration (a) over the planning horizon. For efficiency, sampling step during generation, and T-Time, the total computation time to generate an entire path. Formal definitions of the metrics provided in Appendix E.2
report S-Time, the computation time per
4.1 MAIN RESULTS ON MAZE2D NAVIGATION
We first present the main performance comparison the Maze2D setting, as shown in Figure where there two safety constraints (red circles). Our results illustrate that SafeFlow generates smooth, efficient paths that effectively avoid obstacles, whereas baselines may produce unsafe, suboptimal, computationally-xxeessiee paths.
t=T
t= 0.75T
t= 0.50T
t= 0.25T
0
RES- RES-SafcDifing SafeDifuuerr
(Xiao ct 2025)
SafeFlowMatcher
RLC
(Ours)
1- 0.00
1 0.25
0.50
0.75
1.00
Figure 3: Comparisons of the path generation process in Maze2D. Red circles indicate the safety constraints the path should satisfy. (Top) DES-SafeDiffuser initializes samples all over the maze and converges to path that has local traps. (Bottom) SafeFlowMatchhr (ours) initializes from near target path after prediction phase, and converges higher-quality path with no local traps!
As shown in Table [ SafeFlowMatcher achieves the highest score while preserving safety, with almost local traps. Local traps remain rare even with far more than two constraints: Appendix F.I shows the case with six constraints.
that our method consistently outperforms all baselines, both safety- enabled and disabbed versions, across all sampling horizons. As detailed in Appendix F.3 regarding
Moreover,
Diifuusionbbsedd samplers evolve backward interval |0. 7]. whercas flow matching evolves forward (0, 1}: a natural correspondence can he established by normalizing and reversing time.

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both safety and efficiency, our method also maintains 100% safety even at very short sampling hori- zons. Especially, when 7 4 and using the closed-form solution, our method achieves 50x faster T-Time than SafeDiffuser (0.023s 1.208s), while SafeDiffuser still suffers from severe local traps that lead to incomplete paths. When using the QP solver, SafeDiffuser completes generation in 9. 9.998s, whereas SafeFlowMatcher completes generation in just 0.157s. Notably, the QP-based Safe- Flowhatcher is still about 8x faster than even the closed-form version of SafeDiffuser (1.208s), while achieving high task performance. We further analyze the distributional drift introduced by CBF-based corrections in Appendix F.4
BST (1)
B82 (1)
Score (f)
TRAP
(4)
(+)
(2 0)
(2 0)
(ms)
RATE
Diluuee Jannered 2023
0.825
-0.784
1.72+00288
3.70
0%
77.04+4.330
86.6843.81
DDIM 0.00
0.642
-0.902
1444.11.100
3.63
0%
64.5111.355
57,46.12.44 46
DDIM 1.0)
00.95
-0.899
-565++0.140
3.72
0%
64.21+5.00
57,15+1 96
FM
1.422+0.359
$2.09:+22.02
33.96:2:2.95
Flow Matchrr
-0324
-0.904
1632-+0.003
3,51
D66
73,51++1.02
88,45+00 60
Trancation (15rccmman al 2016
0.999
01.999
0.78100.1 12.8
19.31
100%
Dhariwal & Nictoo RUMB
-0.96
-0.999
0.505+00.00 092
19.13
100%
949 349.63--111 103.62
959 119971-18466
CG-2 Diaanwa Nicciol 2024
-0.998
-0.999
0.49911.100
19.87
100%
1027 1027,28::114..0
1.20249 1961e100
1.435+0.502
100%
15.1576.67
422
RES SafeDilluser
Xiao 21
2025
0.010
0,010
1.442.0.45
4.72
72%
10.30
398 19-117-0
TVS SafeDifiuser X100
2025
D0003
-0.003
0.5
4,78
69%
78, 7.84.2.
124.51 4.34 22
SalcDDMM
0:00)
100%
SaSs-SacDDDMM) 0.0)
0.010
0.010
1.05=20-194
4,83
96%
43.23
1153.8 2040.99
TVS- SuS-SaiDD 0.0)
-0.026
-0.026
1.522.10.99 295
4.79
90%
42.56.13.39
$75.73.1371.88
ROS-SafeDDDMMG 1.00)
1.575+0.158
100%
56.30+2.93
668.
RES SafeDDIMLy 1.0)
0.010
0.010
1.532.0.331
4,82
86%
61.73:14.80
1584.00.1 808S 06.06
(V 1.0)
-0.026
-0.026
1.549+00.8
4.74
65%
60. 60.29+33.41
27.23-4-43,22
ROS-SufeFP
0.010
0,010
1.13::0.556
4.68
100%
23.57::8.34
13177449.931ch
RES-SafeFM
0.010
0.010
1,40++0422
4,74
12%
61,17+19,52
6724 64++5. 304e4
TVS-SufeFM
60.2911.411
768.71=2212.17
SafeFlow Matcher wio relaxation (ours)
0.010
0.0TO
1622+50.05
4.76
25
71.335334
108-435167,366
SafeFlowMaccher (ours)
0.010
0,010
1.63210.003
4.71
0%
69.1911.02
91,9021.777
Table 1: Performance comparison of different methods. We evaluated all methods over 100 inde- pendent trials under identical settings. For all safety-aware methods, we set the robustness margin to 5 = 0.01, meaning that method is considered safe only if b(7) 2 6. This ensures robust rather than marginal safety. FlowMatcher- -araachher-vvriants use T 1 and 7 256, and others use T 256. The closed-form CBF-QP computation takes 1.14 ms on average. All baselines are reproduced by us.
1.56
1.50
Score
1.45
1.40
addibacuccee (can)
Honkitttar lrand
SdtCaar Blavstal, 2351
20032
22
125
16
32
150
25c
Sampling Borizon
Sampling Horizon
Figure 4: Score versus sampling horizon T. Left (safety on): SafeFlowMaachher attains the highest score across all sampling horizons. Right (safety off): FlowMatcher (FM + PC integrator) also re- mains more efficient than the other cases.
4.2 ABLATION STUDIES ON PC INTEGRATOR
Effect of Using Two Phases. To highlight the necessity of both the prediction and correction phases, we discuss the results of Flowhatcher (prediction-only). SafeFM ccorrection-only and Satcheow (PC integrator) in Table The prediction-only behavior achieves good task per- formance but lacks safety. Conversely, the correction-only behavior enforces safety from the be- ginning but often fails to generate complete paths, resulting in a high trap rate. Matchee combines the strengths of both phases, achieving superior performance while ensuring safety.
Matcher
Effect of Prediction Horizon (TP). We analyze how the prediction horizon TP affects overall performance while keeping the correction horizon fixed 7 256. Table 2 reports the qualities
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of fully generated paths and the total computation time across different values of T", and Figure visualizes how increasing 7 shapes the predicted path before correction. As T increases, the path quality remains largely unchanged, while the computation cost increases due to additional prediction steps.
Table Effect of prediction horizon TP. We compare path quality metrics (score, curvature, and acceleration) and the total computation time, measured after one full path generation.
Prediction horizon (1
Score (f)
1.632:0.008
1.520=0.340
1.468=0.434
1. 1.404:0.538
1.632:0.000
T-TIME (s)
1.209
1.220
1.230
1.249
1.287
Curvature x(L)
69.19-1.02
68 8.33..323
68.70.1..62
68.26:14.77
67.7314.97
Acceleration a(L)
91.90+22.77
93.76+2..30
18
+1.99+3.88
92.61
P
1-
7--
P-
7--8
7 16
Figure 5: Predicted paths under different prediction horizon TP, Each shows the predicted path after the prediction phase. As prediction horizon T increases, prediction error & decreases.
Effect of Vanishing Time-Scale. We first analyze the role of the scaling constant a in VTFD 10) As shown in Table Figure increasing a consistently reduces both curvature and acceleration, indicating that larger scaling factors suppress the prediction error more aggressively. This trend is consistent with the theoretical result from Lemma
However, we observe that larger introduces bias in the final path. This effect is visible in Figure where the red path region stays relatively stable up to a certain critical value but becomes increas- ingly distorted once exceeds this threshold. In our Maze2D setup, this occurs around x 2. This shows that a should not simply be maximized in practice; instead, one can start from a 1 and increase it until we identify the point just before the sharp distortion begins.
Figure 7 shows how the score changes with increasing correction horizon T when T 1. With vanishing time-scale, the score remains stable even T grows, whereas removing the scaling causes the score to deteriorate steadily. Figure provides the corresponding path visualization. With vanishing time-scale, the correction path moves from 7 of along straight direction. In contrast, without scaling, the path exhibits sharp drift near /=1, and some segments of the path become largely distorted. These results demonstrate that vanishing time-scale is essential for preventing late-stage drift and maintaining stable refinement behavior.
Table 3: Effect scaling constant a. Path qualities are measured after full generation.
Scaling constant
1.0
1.5
2.0
2.5
3.0
Score (1)
1.623+0.000
1.629+00.0M
1.632+00.008
1.572+0.058 0.618
Curvature x (1)
85.10+3..73
83.91+2.000
19.28+1.004
55. 55.16+0.83
44.08-+0.62
Acceleration (4)
173.22+5.62
123.49
92.05:+0.59
71.89+0.42
58.05
L
1.0
1-15
2-
2.5
0 3.0
Figure 6: Generated paths under different scaling constant Each snapshot shows the fully generated path after the two phases. As the scaling constant a increases, the path becomes smoother but can be distorted.

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Score (1)
vanishing lim-scale we vantshing time scale
1.0
0.8
(b)
256
Correction Horizon T
Figure 8: Generation process with (a) and with- out (b) vanishing time-scale. The transparent path represents T the solid path represents Tf- The black line represents the path oi from over the interval E [0, 1]. The path's segments in the black circles are largely distorted in the ab- sence of vanishing time-scale.
Figure 7: Score with and without a vanish- ing time-scale. When 7 as the correction horizon 7 increases, we see that the score de- creases in the absence of vanishing time-scale.
4.3 GENERALIZATION TO HIGH-DIMENSIONAI ROBOTIC TASKS
We evaluate the generalization capability of SafcFlowMatcher on high-dimensional robotic tasks, including two locomotion environments (Walker2D and Hopper) and robot manipulation task (Block Stacking). Across all three tasks, SafehlowMatcher attains the highest score while main- taining BS 0, indicating that the PC integrator scales beyond static maze navigation. The detailed comparison across locomotion and manipulation tasks is summarized in Table 4 Note that the BS metric here is reported in different way than in Table I here, BS binary indicator (yes or no) of whether safety is guaranteed (2 0) or not (< 0).
Table 4: Performance on high- high-ddmensioonl robotic tasks. SafeFlow.Matcher maintains its advan- tages in both locomotion and robot manipulation settings.
Category
Environment
Method
Score (f)
BS (> 0)
SafeDiffuser Xiao et al. 2025 SateFM
0.283 : 0.135
Yes
Walker2D
0.264 0.127
Ours
0.331 0.021
Locomotion
SafeDitluser Xiao et al, 2025
0.435 0.065
Yes
Hopper
SatcEM
0
Yes
Ours
0.917
SafcDiftuser (Xiao et 2025
Robot Manipulation
Block Stacking
SaleF
0.73 3-73+100 0.056
Yes
Ours
0.76 1:: 0.053
Yes
5 CONCLUSION
We introduced SafeFloowatther: a planning framework that couples flow matching (FM) with CBF- certified safety by employing a two-phase crrrectoon-oorrection integrator. On the path generation side, we proposed the vanishing time- scaled flow dynamics, which contracts the prediction error to- ward the target path. On the safety side, we established finite-time convergence barrier certificate for the flow system to ensure forward invariance of safe set. The approach generates candidate path with the learned FM dynamics and then refines only the executed path under safety constraints. This decoupling preserves the native generative dynamics, avoids distributional drift from repeated interventions, and mitigates local trap failures near constraint boundaries. Empirically, SafeFlow- Matcher attains faster, smoother, and safer paths than various diffusion- and FM-based basclines across maze navigation, locomotion, and robot manipulation tasks. Incorporating data-driven cer- tificates is promising direction for extending certified generative planning more dynamic and complex environments.
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ACKNOWLEDGMENTS
We would like to thank Jiwon Park for assistance with setting up the initial experiments.
This research was supported by Brain Pool program funded by the Ministry of Science and ICT through the National Research Foundation of Korea(RS-20255 2025-2544348..
REPRODUCIBILITY STATEMENT
All baseline results reported in this paper are fully reproduced by using mentations or publicly available code, ensuring a fair and controlled comparison on the same hardware. To facilitate reproducibility, evaluation the supplementary material. For fair comparisons under matched computational budgets, our model architectures strictly adhere those in prior work (Janner al., [2022] Xiao et al. 2025) and their official implementations (Code: Our experiments are conducted on the Maze2D environ-
imple-
provide anonymized source code for training and
https://gituu..com /aa///eiil//Sfeefffuussee ment, locomotion tasks (Hopper, Walker2D) and a robot manipulation task (block stacking). All hyperparameters for training and evaluation, including optimizer settings, learning rates, and rollout configurations, are detailed in Appendix [ For each experimental setting, we perform 100 inde- pendent trials and report the mean and standard deviation across these runs in Table [ and Table [ All experiments were run on machine equipped with an AMD EPYC9354 CPU and NVIDIA RTX4090 (24GB) GPU. Additional ablation studies are provided in Appendix F
REFERENCES
Hossein Abdi, Golnaz Raja, and Reza Ghabcheloo. Safe control using vision-basee control barrier function (V-CBF). In IEEE International Conference on Roboties and Automation (ICRA), pp- 782-788. IEEE, 2023.
Ayush Agrawal and Koushil Sreenath. Discrete control barrier functions for safety-criticd control of discrete systems with application to bipedal robot navigation. In Robotics: Science and Systems, volume 13, pp. 1-10. Cambridge, MA, USA, 2017.
Aaron D Ames, Xiangru Xu, Jessy W Grizzle, and Paulo Tabuada. Control barrier function based quadratic programs for safety critical systems. IEEE Transactions on Automatic Control, 62(8): 3861-3876, 2016.
Aaron D Ames, Samuel Coogan, Magnus Egerstedt, Gennaro Notomista, Koushil Sreenath, and Paulo Tabuada. Control barrier functions: Theory and applications. In 2019 18th European control conference (ECC), pp. 3420-3431. IERE, 2019.
Sanjay P Bhat and Dennis S Bernstein. Finite-time stability of continuous autonomous systems.
SIAM Journal on Control and optimization. 38(3)7751-766, 2000.
Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004.
Max Braun, Noémie Jaquier, Leonel Rozo, and Tamim Asfour. Riemannian flow matching policy for robot motion learning. In 2024 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 5144-5151. IEEE, 2024.
Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. OpenAl Gym. arXiv preprint arXiv: (60:::600011500 1540, 2016.
Joao Carvalho, An T Le, Mark Baierl, Dorothea Koert, and Jan Peters. Motion Planning Diffusion: Learning and planning of robot motions with diffusion models. In 2023 JEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 1916-1923. JEEE, 2023.
Chang Chen, Fei Deng, Kenji Kawaguchi, Caglar Gulcehre, and Sungjin Ahn. Simple hierarchical planning with diffusion. In The Twelfth International Conference on Learning Representations, 2024.
11

---

Published as a conference paper at ICLR 2026
Eugenio Chisari, Nick Heppert, Max Argus, Tim Welschehold, Thomas Brox, and Abhinav Valada. Learning robotic manipulation policies from point clouds with conditional flow matching. In 8th Annual Conference on Robot Learning, 2024.
Ryan Cosner, Andrew W Singletary, Andrew Taylor, Tamas Molnar, Katherine Bouman, and Aaron D Ames. Mcasurement-robust control barrier functions: Certainty in safety with un- certainty in state. In 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. f286-6291. IEEE, 2021.
Prafulla Dhariwal and Alexander Nichol. Diffusion models beat GANs on image synthesis. In Advances in neural information processing systems, volume 34, pp. 8780-8794 0794,,874444 2021.
Ruiqi Feng, Chenglei Yu, Wenhao Deng, Peiyan Hu, and Tailin Wu. On the guidance of flow matching. In Proceedings of the 42nd International Conference Machine Learning, volume 267, pp. 16993-17029. PMLR, 2025.
Marvin Harms, Mihir Kulkarni, Nikhil Khedekar, Martin Jacquet, and Kostas Alexis. Neural control barrier functions for safe navigation. In 2024 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp- (0415-10422, IEEE, 2024.
Jonathan Ho and Tim Salimans. Classifier-free diffuusin guidance. In NewrlPS 2021 Workshop on Deep Generative Models and Downstream Applications, 2021.
Jonathan Ho, Ajay Jain, and Picter Abbecl. Denoising diffusson probabilistic models. In Advances in neural information processing systems, volume 33, pp. 6840-6851, 2020.
Kai-Chich Hsu, Haimin Hu, and Jaime F Fisac. The safety filter: A unified view safety-critical control in autonomous systems. Annual Review of Control, Robotics, and Autonomous Systems, 7, 2023.
Michael Janner, Yilun Du, Joshua Tenenbaum, and Sergey Levine. Planning with diffusion for ficxible behavior synthesis. In Proceedings of the 39th International Conference on Machine Learning, volume 162, pp. 9902-9915. PMLR, 2022.
Zhuozhu Jian, Zihong Yan, Xuanang Lei, Zihong Lu, Bin Lan, Xueqian Wang. and Bin Liang. Dynamic control barrier dunetion-based model predictive control to safety-critical obstacle- avoidance of mobile robot. In 2023 IEEE International Conference on Robotics and Automation (ICRA), pp. 3679-3683 IEEE, 2023.
Hassan K Khalil and Jessy W Grizzle. Nonlinear systems, volume 3. Prentice hall Upper Saddle River, NJ, 2002.
Jeeseop Kim, Jaemin Lee, and Aaron D Ames. Safety-critical coordination for cooperative legged locomotion via control barrier functions. In 2023 IEEE/RSJ International Conference on Intelli- gent Robots and Systems (IROS), pp. 1368-2375... IEEE, 2023.
Anqi Li, Li Wang, Pietro Pierpaoli, and Magnus Egerstedt. Formally correct composition of coor- dinated behaviors using control barrier certificates. In 2018 IEEE/RSJ International Conference Intelligent Robots and Systems (IROS), pp. 3723 3729. IEEE, 2018
Qiayuan Liao, Zhongyu Li, Akshay Thirugnanam, Jun Zeng, and Koushil Sreenath. Walking in narrow spaces: Safety-critical locomotion control for quadrupedal robots with duality- opti- mization. In 2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IRÓS), pp. 2723-2730. IEBE, 2023.
Lars Lindemann, Alexander Robey, Lejun Jiang, Satyajeet Das, Stephen Tu, and Nikolai Matni. Learning robust output control barrier functions from safe expert demonstrations. IEEE Open Journal of Control Systems, 8:155-172, 2024.
Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. In The Eleventh International Conference on Learning Repre- sentations, 2023.
12

---

Published as a conference paper at ICLR 2026
Jianwei Liu, Maria Stamatopoulou, and Dimitrios Kanoulas. Dipper: Diffusion-based 2d path plan- ner applied on legged robots. In 2024 IEEE International Conference on Robotics and Automation (ICRA), pp. 2264-9270 IEEE, 2024.
Luping Liu, Yi Ren, Zhijie Lin, and Zhou Zhao. Pseudo numerical methods for diffusion models on manifolds. In International Conference on Learning Representations, 2022.
Kehan Long, Cheng Qian, Jorge Cortés, and Nikolay Atanasov. Learning barrier functions with memory for robust safe navigation. IEEE Robotics and Automation Letters, 6(3):99314998,, 2021.
Kehan Long, Vikas Dhiman, Melvin Leok, Jorge Cortés, and Nikolay Atanasov. Safe control syn- thesis with uncertain dynamics and constraints. IEEE Robotics and Automation Letters, 7(3): 7305-7302, 2022.
Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. DPM-Solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps. In Advances in neural information processing systems, volume 35, pp. 5775-5787, 2022.
David G Luenberger. Optimization by vector space methods. John Wiley & Sons, 1997.
Mitio Nagumo. Über die lage der integralkurven gewöhnlicher differentillleechunggn. In Pro- ceedings of the physico-mathematica society of Japan. 3rd Series, volume 24, pp. 551-559. The Physical Society of Japan, The Mathematical Society Japan, 1942.
Ahmed H Qureshi, Anthony Simeonov, Mayur Bency, and Michael C Yip. Motion planning net- works. In 2019 International Conference on Robotics and Automation (ICRA), pp. 2118-2124. IEEE, 2019.
Alexander Robey, Haimin Hu, Lars Lindemann, Hanwen Zhang, Dimos V Dimarogonas, Stephen Tu, and Nikolai Matni. Learning control barrier functions from expert demonstrations. 2020 59th IEEE Conference on Decision and Control (CDC), pp. 3717-3724. IEEE, 2020.
Jiaming Song. Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. In Interna- tional Conference on Learning Representations, 2021a.
Yang Song, Jascha Sohl-Dickstein, Diederik Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In Interna- fional Conference on Learning Representations, 2021b.
Mohit Srinivasan, Samuel Coogan, and Magnus Egerstedt. Control of multi-agent systems with finite time control barrier certificates and temporal logic. In 2018 IEEE Conference Decision and Control (CDC), pp. 1991-1996. IEEE, 2018
Mohit Srinivasan, Amogh Dabholkar, Samuel Coogan, and Patricio A Vela. Synthesis of control barrier functions using supervised machine learning approach. In 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 7139-7145. IEEE, 2020.
Emanuel Todorov, Tom Erez, and Yuval Tassa. MuJoCo: A physics engine for model-based control. In 2012 JEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 5026-5033.
IEEE, 2012.
Kim Wabersich, Andrew J. Taylor, Jason J. Choi, Koushil Sreenath, Claire J. Tomlin, Aaron D. Ames, and Melanie N. Zeilinger. Data-driven safety filters: Hamilton-Jacohi reachability, control barrier functions, and predictive methods for uncertain systems. IEEE Control Systems Magazine, 4375):137-177, 2023.
Li Wang, Aaron D Ames, and Magnus Egerstedt. Safety barrier certificates for collisions-fre free mul- tirobot systems. IEEE Transactions on Robotics, 13(6),6616677, 2017.
Wei Xiao, Tsun-Hsuan Wang, Chuang Gan, Ramin Hasani, Mathias Lechner, and Daniela Rus. SaffDiffuser: Safe planning with diffusion probabilistic models. In The Thirteenth International Conference Learning Representations, 2025.
13

---

Published as a conference paper at ICLR 2026
Zebin Xing. Xingyu Zhang, Yang Hu, Bo Jiang, Tong He, Qian Zhang, Xiaoxiao Long, and Wei Yin. GoalFlow: Goal-driven flow matching for multimodal trajectories generation end-to- end autonomous driving. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition (CVPR), pp. 1602-1611, June 2025.
Brian Yang, Huangyuan Su, Nikolaos Gkanatsios, Tsung- Wei Ke, Ayush Jain, Jeff Schneider, and Katerina Fragkiadaki. Diffusion-ES: Gradient-free planning with diffusion for autonomous and instruction sectecttoonguidedd guided driving. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 15342-15353, 2024.
Sean Ye and Matthew Gombolay. Efficient trajectory forecasting and generation with conditional flow matching. In 2024 IEEE/RSJ International Conference Intelligent Robots and Systems (IROS), pp. 2816-2823. IEEE, 2024.
Fan Zhang and Michael Gienger. Robot manipulation with flow matching. In CoRL 2024 Workshop on Mastering Robot Manipulation in World of Abundant Data, 2024.
Qinsheng Zhang and Yongxin Chen. Fast sampling of diffusion models with exponential integrator. In NeurlPS 2022 Workshop on Score-Based Methods, 2022.
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A ADDITIONAL RELATED WORK ON CONTROL BARRIER FUNCTIONS
Control Barrier Functions have been developed and extended in a wide range of directions, and existing results show that CBF-based safety filter does not rely on perfectly known, smooth, or analytically specified safe sets, Discrete-time CBFs have been applied to hybrid locomotion and time-varying safety constraints (Agrawal & Sreenath 2017). and duality-based DCBF methods able safe control even with nonsmooth, polytopic, or nonconvex obstacle geometries (Liao et al.) 2023). Perception noise and state-estimation uncertainty can be handled using measurement-robust and probabilistic CBF formulations (Cosner et al. 2021 Long et al. (022). Moreover, CBFs have been extended to dynamic-obstacce environments, explicitly incorporating obstacle motion predic- tion and enabling real-time avoidance of moving obstacles (Jian et al. )023).
In addition analytic formulations, growing line of work develops bearning-based CBFs that construct safety certificates directly from data rather than hand-crafted functions. These methods learn barrier functions from RGB-D observations (Abdi al. 2023). LiDAR scans (Srinivasan et al. 2020 Long et al., 202 Harms et al. 2024). or expert demonstrations (Robey et al. 2020 Lindemann et al. 2024D, enabling implicit representations of safe sets in dynamic and unstructured
environments. W While SafeFlow currently leverages analytic CBFs, its correction phase only requires evaluating a barrier constraint, making the framework compatible with these learned or pereeption-driver CBFs.
Beyond their theoretical development, CBF-based safety filters have also been applied across a wide range of robotic domains. They have seen successful use in autonomous driving (Ames al. 2016), legged locomotion (Kim et al. 2023), and multi-robot coordination (Wang et al. 2017).
B THEORETICAL AND EMPIRICAL SUPPORT FOR THE CORRECTION PHASE
B. EMPIRICAL VALIDATION OF THE PREDICTION ERROR ASSUMPTION IN LEMMA 2
Lemma 2 and Lemma 3 assume that the prediction error & follows symmetric, zero- mean distribu- tion in a neighborhood around the target path. We empirically validate this assumption by evaluating the distribution of e under different prediction horizons T E {1, 4, 8, 16. 32} in the Maze2D en- vironment. For each configuration, we generate 1,000 predicted paths, resulting in total of 384.000 waypoints, and evaluated the prediction error with respect to a high-accuracy FM solution TJ. which is computed using the Dormand- Barmand-Prince 5(4) method (Dopri5) with 256 steps.
Figure 9 visualizes results. Across all values 7", the distribution of remains centered at zero and exhibits symmetry, directly supporting the symmetric zero-mean (Gaussian-like) assumption used in both lemmas. Validating whether this assumption still holds for nigher-dimensional, complex tasks is a subject of future work. However, we anticipate that while the final refined path may be biased if the prediction error is biased, overall safety is still unaffected because the CBF-QP enforces forward invariance regardless of any bias. Moreover, if the bias is heavy-tailed, the local strong convexity of log Ps becomes weaker, which may slow down the contraction rate in the correction phase. Again, this only affects path refinement speed, not safety guarantees, and increasing a to introduce deliberate path distortion against the error (see Table and Figure might help the prediction error reduction.
B.2 PROOFS OF LEMMA 2 AND LEMMA 3
Proof of Lemma 2 Let (c) = Ti + 85, where 5 A 1 t. We have pushforward of Pe under 61:
TØb 0T (T)
1
( (--7)
(T TI) (E) p= (T)) det
8a(n+1)PP
(8
By Bayes' rule,
Tf TI
p( T() x pı (Ti Pe
Since log Ps (2) is C near 0 with Hessian A 0 by the assumption,
(=)))
1 TT Az + O(1==19).
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Let y - T1 Tf. Substituting y/ yields the posterior energy
(5(y)
1 25a y Ay Po + v) O(1).
The quadratic term dominates as 6 (t + 1). so the posterior concentrates an O()) neighbor- hood of T.
The stationarity condition ( gives
1
8² Ay V log (T + v) 0.
Taylor expanding V llogpo pı at oi shows = O Thus the posterior mode
i = _ 8² V
+ O(5*).
Laplace's 's approximation then yields the same expansion for the posterior mean:
T 7 = -. 82 A lo (7() + 0
Under the assumption, we have 7r 7 + de with ||||| O()),
E Ti Tf = - + 6s + 0 = = O(6).
This proves Lemma 2
Proof of Lemma the flow dynamies follow the vanishing time-scaled flow dynamics (1O). then we have:
+ a(1 (T(T:: 0) (E|T -
Let e A Ti Ti E ( and denote its k-th column by ex,t E Rd. By Lemma Tff -ं +0 1) 1, hence have
E [T)
d = ack, 0(1 - t).
Solving with an integrating factor gives
Cket -at ek ck,o + ae S(" - (00 (ex.o + 0(1))e + +0((1 (((1+))3
Combining the column vectors again yields the form
(eo +0())) O(1 - 1)²):
eo =.
which proves Lemma
C PROOF OF THBOREM 1 AND PROPOSITION
We drop the superscript e for simplicity, and choose the Lyapunov candidate function V(T A max(5 b(T(),0). Since w(l) = for 2 bu the barrier inequality (15) reduces on [Lm, to
b(T+))
(g(()((T)) - 6) [b(T.t 510 2 0.
Case If The E Cs (i.e., b(T 2 5), then V(T O. For all 2 two b(T!") 5 we have
V(T(") = O. If ((T(t) = 8, the barrier inequality 15 with (g) = 0 reduces to b(T()
) 2 0, so the 1942 Therefore V(T²*) for all > tw.
path cannot exit Cs by Nagumo's principle (Nagumo which implies Tha E Cs: the system stays in Cs.
Case 2: T/ & (i.e., b(T!).) 8). then ((T 8 b(7,t) 0. The following staite-stability condition holds
((TT)) = -((T)) < -(6 b(TT))) - (T/))
Nagumo's 's theorem states that if the vector field at the boundary lies in the tangent cone of set, then the set forward invariant.
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7
6.5
0.03
-1.6
K.66 100
-0.15
-2225 10
2.343x 10
-1.15
5.418 1
8.7.00 10
1.342 10-1
6.460x100 10-
-0.5
6.0
0.
-0.23
-0.5
3.4
7P
6.0
110
0.03
0.00
2.9979
3.034 x30
-0.15
6..004 10
33.125
-1.15
5.435 K.10
-3.167
9.002 10
8.023 x1-02
-0.3
6.00
-0.23
-0.35
030
TP 16
6.3
10
0.03
3.00
3.70xx10
1.252
6.15
4.530 10
-2244x10
1.15
-3.353
x.00
7,266 x 10-
1.367 10
-0.33
680
-0.33
-0.25
Figure 9: Empirical distribution of the prediction error over prediction horizon TP, The six subfigures correspond to T = 1, 4, 8, 16, (from top-left to bottom-right). Each subplot visualizes the joint density of (ε,,E,) with its marginal distributions. As Tp increases, the error distribution becomes more concentrated around zero while maintaining symmetry, validating the symmetric zero-meat mean assumption used in Lemma 2 and Lemma
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Define the comparison system
o(t) -CO(t), x(tw) V(T...).
By the Comparison Lemma (See Lemma 3.4 in Khalil & Grizzle 2002 ) we have:
V(T/") - (t). Vt 2 bw
The solution (()) is
(bt) (V (V(T)))) (1 (1-p)e(t tw:) 1 fort tw.
Thus,
v(T S (V(T)-) - (1- p)e(t tw)) -ーp
Hence, 7² reaches the robust safe set Cs in finite time II' that satisfies V(T,') S o(T) = 0. Moreover, we get the finite convergence time,
V )T- = tw (6 ((TT(,))
T = tw
E(1 p)
c(1 p)
Therefore, for all f. 2 7, we have ((T s 0, implying x E Ca. This completes the proofs of both Theorem [ and Proposition
D DIFFERENCES IN LOCAL TRAP DEFINITIONS
We clarify the difference between the local trap definition used in our SafeFlowMatchnn Matcher and that of the baseline method SafeD- iffuser (Xiao et 2025).
SLL
Definition (Local Trap in SafeDiffuser) local trap problem occurs during the plamning process if there exists h H such that b(T()=0 and || T| Ç. where {> O is user-defined threshold depending on the planning environment.
In contrast, our definition of a local trap in SafehlowMatcher re- moves the condition b(rr") 0 and instead considers only the relaxing the con-
abrupt discontinuity in the path. The reason dition is illustrated in Figure In this example, the generated path incomplete due to overly strong or early intervention of the CBF However, since the waypoints do not strietly lic the boundary (i.e., b(T")) ) * 0), the original SafeDiffuser definition fails to detect this failure local trap. Therefore, we generalize the definition capture wider class of failure cases.
Figure 10: Local trap occur- ring away from the safety boundary. Although some waypoints do not violate con- straints (i.e., b(T²))
0), it fails to reach the goal. Our definition considers such cases as local traps, while the original definition does not.
E EXPERIMENTAL DETAILS
E. EXPERIMENTAL SETUP
All CBF constraints are enforced via the closed-form projection of the CBF-QP in (17). For each model family, the safety-enabled variants reuse the same trained weights as their safety-disabled counterparts. Specifically, SafeDiffuser and SafeDDIM share the weights trained for Diffuser and DDIM, respectively, while SafeFM and SafeFlow.Matcher share the weights trained for FM and FlowMatcher. All experiments are run using an AMD EPYC 9354 CPU and an NVIDIA RTX 4090 GPU (24GB).
To match the total amount training data used in Diffuser (Janner et al. 2022). we first swept across several batch sizes while fixing the total number of samples processed during training to 6.4 x 10?. As shown Table 5 both Diffuser and FM performed best or on par at batch size 1 128, for all models and Maze2D experiments, used batch size 128. Other training
Maze2D.
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and inference hyperparameters shown in Tables 6 For the correction phase, we set the scaling constant to a 2. and use (ó, p) (0.01, 0.5, 0.9) for the CBF parameters. Additionally, for the relaxation schedule, tu is chosen according to the correction horizon T". Specifically, tn E (0, 0.5, 0.75, 0.9, 0.9, 0.9, 0.99} for 7 ε (4,8, 16, 32, 64, 128, 256), respectively. The relax- ation function is defined as w = 200(1 - 23t// 1) for I two and w = otherwise. For Maze2D, the planner is conditioned on the start and goal observations, which provided as the condition for each rollout.
Table Scores by batch size for Maze2I for both Diffuser and FM.
Method FlowMatcher
16
32
64
128
256
1661
0.003
1.628 + 0.002
1.615 + 0.031
1.631 = 0.003
1.523 = 0.196
Diffuser Janner et al. 2022
1.503
0.424
1.438 + 0.500
1.516 0.316
1.537
0.265
1.536
0.338
Locomotion. Following the observations from Maze2D, we also train all locomotion models us- ing a batch size of 128. Safcheoo SafeFM. and SafeDiffuser share the same hyperparameter settings, summarized in Table To provide score-based guidance to all fow-matching- based meth- ods, including SafeFM and Sathelow we apply a simple covariance-awar aware guidance goor- with scale 1.0, following prior work (Feng et al. (025). During planning, we condition the model at each environment step on the current state observation and use the task score as guidance signal to encourage forward progress.
(a) Walker2D planning result with Matcherw
(b) Hopper planning result with SafcFlow
Figure 11: Safeelow.Matcher on locomotion tasks. Planning results for Walker2D (top) and Hop- per (bottom). In both figures, the red horizontal line indicates the roof height h, in the CBF barrier function (z < h,y) used in the BS metric (Appendix))
Robot Manipulation (Block-Stacking). For the block stacking task, we followed the training parameters from Diffuser Jaanerr et al.] (2022) (batch size 32 with 2-step gradient accumulation, equivalent to batch size 64 without accumulation), rather than using batch size of 128, while maintaining the number training steps for training SafeFllowMatchher(SafaFM sharing weights with Safeeloowaatcher and SafeDiffuser. Other hyperparameter values and conditions are shown in Table 8 For the block stacking task, the condition includes the initial robot joint configuration together with the observed states of the four blocks.
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(a) = - 999
(b) T 600 400
Figure 12: Block stacking visual comparison based on the prediction and correction horizons. For each subfigure, the left shows the predicted path 7f and the right shows the corrected path rf- Under the same planning horizon
128, we compare different allocations of T and 7c. In (a). using 7 1 and 7 999 leads poor prediction quality due the short prediction phase, resulting in large prediction error and ultimately a failed path. In contrast, (b) uses 7 = 600 and and successfully produces safe and complete path, where the yellow block is being stacked on top of the green block.
7 400 which yields small prediction
Table 6: Maze2D's training and evaluation hyperparameters
Training
Loss type
L2
Training steps Mhmin
5.0 x 105
Steps per epoch
2500
Batch size
128
Learning rate
10-4
EMA decay
0.995
Evaluation Others
Planning Horizon
384
Sampling Horizon T
256
Evaluation SafeFlowMMatcher
Planning Horizon H
384
Prediction Horizon TP
Correction Horizon T
256
E.2 PERFORMANCE METRICS
BS quantifies the degree of safety constraint satisfaction using CBFs for each safety constraint in the environment. For each rollout, we evaluate the minimum barrier value over all waypoints, and then take the worst case across all N test episodes:
min
min b(T
1.2......
KEH
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Table Locomotion (Walker2D/Hopperr) hyperparameters
Training
Loss type
Training steps Rirain
2.5 x 10a
Steps per epoch
2500
Batch size
128
Learning rate
2 x 10
EMA decay
0.995
Value Network Training
Loss type
L,2
Training steps Thrain
5.0 x 104
Steps per epoch
2500
Batch size
128
Learning rate
10
EMA decay
0.995
Evaluation Others
Planning Horizon H
600
Sampling Horizon T
20
Evaluation SafeFlowwMtccer
Planning Horizon H
600
Prediction Horizon T
Correction Horizon T
20
Table 8: Robot manipulation task (block stacking) hyperparameters
Training
Loss type
I.2
Training steps Tirain
7.0 x 105
Batch size
64
Learning rate
10
EMA decay
0.995
Evaluation Others
Planning Horizon H
128
Sampling Horizon T
1000
Evaluation LafeFlooMMatchee
Planning Horizon H
128
Prediction Horizon TP
600
Correction Horizon T
400
value BS > indicates that the path remains entirely within the safe set. Maze2D contains two obstacle-based safety constraints, given by the barrier functions:
2
20
y0
("
y0
BS1
+
BS2
+
a
a
b
where (x, y) E R22 denotes the agent's ' 2D state, (xo. 3o) E R² specifies the center of the obstacle, and a, 6 0 are scaling parameters that shape the corresponding safety region. For locomotion tasks (Walker2D, Hopper), the barrier function is defined as 2 & hy, where h, > denotes the roof height. For robot manipulation task (block stacking), the safety constraints enforce joint limits. The barrier functions are defined as qmin S $ 9max, where E R7 denotes the joint-angle and Gmin, Gmax RR7 are the per-joint limits.
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Score is normalized, undiscounted performance metric that reflects task success. In Maze2D, episodes last up to 800 environment steps while planning is performed over a horizon of = 384; once the agent enters goal neighborhood, it receives unit reward for each remaining step, making the score proportional to the remaining horizon. For locomotion tasks, the score is proportional to forward displacement and normalized such that reaching the target position manner, continuing until the agent either reaches x= 1, falls, or reaches the maximum episode limit of 1000 steps. For robot manipulation task (block stacking), planning is performed with horizon H = 128, and each episode attempts a single stacking action. An episode receives a score of upon a successful stack and 0 otherwise.
yields score of 1.
We evaluate locomotion in a receding-horiza
Trap Rate measures the rate of local traps, i.e., the percentage of episodes in which the generated path becomes stuck against barrier constraints; see Definition 2
T-Time & S-Time. We report two timing metrics: the total computation time (T-Time) and the per- step sampling time (S-Time), both of which include all computational overheads such as CBF-QP evaluations. T-Time measures the total wall-clock time required to generate an entire path, including all prediction and correction steps when applicable. S-Time reports the average wall-clock time per sampling step, computed as S-Time T-Time/TT where T is the total sampling horizon.
Curvature (I:) measures local path bending using the Menger curvature computed from triplets of consecutive points. We report the average curvature along the path.
Acceleration (a) captures the change in velocity across consecutive time steps. is computed as the mean squared acceleration magnitude along the path. We approximate it via the second-order finite difference of the 2D position and define the metric as the average acceleration magnitude along the path.
F ADDITIONAL. ABLATION STUDIES
F.1 HANDLING MULTIPLE CBF CONSTRAINTS AND MITIGATING COMPUTATION BOTTLENECK
We considered only two CBF constraints so far. When more than two constraints are present, no closed-fome solution is available, and QP solver must be used to compute the CBF-QP at every step. This inevitably increases computational overhead and can become a bottleneck.
Figure 13 presents the path generation results with six CBF constraints, under fixed sampling horizon T 256, while varying the allocation between prediction and correction horizon (TP,To). The corresponding T-TIME and Trap Rate for each configuration are also reported. We observe that SafeFlowMMatcher maintains a trap rate of 0% across all settings. However, as T increases, the T-TIME grows due to repeated CBF-QP solves during the correction phase.
A key advantage of Safeelowaatchher is that the PC integrator naturally mitigates this computational bottleneck. Since CBF-QP computations occur only in the correction phase, TP and T can be adjusted to reduce the number of QP evaluations while maintaining safety, In contrast, SafeDiffuser and SafeFM require CBF-QP computations at every generation step, resulting significantly higher overhead when many constraints are present. Moreover, SafeDiffuser becomes unstable in high- constraint settings. As presented in Table local traps are already problematic in the two constraints setting. They occur even more frequently as the number of safety constraints increases. In the six CBF constraints setting, SafeDiffuser required T-TIME = 10.269 and exhibited a 100% trap rate over 100 runs. These observations highlight that the PC integrator enables SafeFlow to scale efficiently and robustly to environments with many CBF constraints, both in terms of computational latency and safe path generation.
F.2 EXPLORING THE FEASIBLE RANGE OF CBF HYPERPARAMETERS AND
We examine the sensitivity of Safeelow.Matcher and SafeFM to the CBF hyperparameters p and c. Smaller p or larger induce more aggressive safety corrections, which can help fast convergence to a safe set but may also increase curvature. When excessively strong, these corrections can even lead unstable oscillatory behavior.
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FLR
FD
FRR
L
T 32, T 224
T 64, T 192
TP 96, T 160 T-IMM: 6.671s. TRAP: 0%
T 128, T 128
T-TIME: 9.025s, TRAP: 0%
T-TIM: 7,913s, TRAP: 0%
T-TMM: 5,556s, TRAP: 0%
R9
S
S
R
T 160, T ===0 90 T-TIME: 4.384s, TRAP: 0%
T" 192, T = 64
T 224, T = 32 T-TIME: 2.070s. TRAP: 0%
T 240, T = 16
T-TIME: 3.222s, TRAP: 0%
T.TIME: 1.473, TRAP: 0%
Figure 13: Balancing prediction and correction horizon in narrow-corridor setting, Visualiza- tion of the prediction-correctio trade-off under a fixed total sampling horizon T TV + T 256. Each result shows the resulting path for different allocation of prediction steps T and correction steps T in the narrow-corridor setting.
Across sweep of E (0.1, 0.5, 0.7, and CE (0.25, 0.50, 1.00, 2.50, 5.00, 10.00), Safe- Flohhr, which include PC integrator, remains stable over a significantly broader hyperpa- rameter range than SafeFM (the correction-only variant), making it substantially easier to tune in practice. This behavior is consistent with Remark [ which explains that the prediction phase places the path closer to a region where safety enforcement is feasible and well- collicondiiinnd,, resulting in more robust behavior under different CBF strengths.
Table 9: Comparison between SaeeFlowMatcher and SafeFM on CBF hyperparameters. Sub- set of the (p. e) hyperparameter grid in Maze2D, comparing Sawcher (ours) and SafeFM (w/o PC). Each entry reports mean + std over 100 rollouts for Score, Trap Rate, curvature (k), ac- celeration (a), and minimum barrier values (BS1, BS2).
Score
Rate
1181
Okess
Voln PC
Oan WC
Oare
wib PC
Dan
G/i PC
Cues
Win PC
Ckm
wil PC
0-445
1:788
94.579 1.116
1.08oodi
0.000
0.002
100
0.691
1047%
78 720
2.375
102.
3.267
2.530e-0
4. 4.300-Da
0.00
-0.41
0.00
0.1
2.50
1633
4 11
0.709
0.699
164
1047%
79 33333-4
1.598
3,041 0.413
169.305
0 5,840
1.750-b0
à 1005p-b4
0.6D
-0.13
0100
0.383
0.1
3.00
1,633
01004
1,025
0.61
264
10472
616
1.505
3.871 0-056
141. 421
1.4
0.00
-0.111
0.00
6.1
10.00
0.003
1.215
0.539
90%
1095
5,147 0.551
174.208
51.12
11500-04
2497o-03
0,000
.0.55
0.010
0.3
0.25
1852
0.638
0.645
166
100%
15,121
1 1.335
3.306 1 0.334
92.501 0.7575
1.555=04
4.55 4555
0.00
-0.01
0.024
0.014
0.3
11:00
653
11.004
0.702
0.737
164
1067
75. 03.438-4
1.367
2.765 = 0.356
93,140 61 11740
1.80 04
1.500-04
0.60
.0056
0.09
0.017
0.3
1.00
1.032
11003
0,412
0.600
6/
1047
77:971
1.577
4.105 0.561
93.835
4.500- 0M à 11260-05
0.00
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0109
0.021
7.836
1e35
3.715-tu
0.00
0109
0.003 1833
3.655 113.198:2
63
10.00
0.003
1.225-
0.416
66
1007
70.345
1.796
21.111 4.519
9.455
3.000e-00
1.490-05
0.00
-0.40
0.00
0.071
0.5
0.25
1,622
2: 11003
1083 0.551
10665
10.226
1.101
4.180 =2 0.607
92. 02.006 0.797
1,55e (M
3,452o 02
0.099
00005
0.05
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110077
1.315
1966
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1.404
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61.00
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91.854 0.76
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92.
1.115
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0.00
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F.3 EFFICIENCY OF SAFEFLOW ACROSS CORRECTION HORIZONS
Table 10: Closed-Form CBF Solver: Computation time across correction horizons. Comparison of RES-SueeD with fixed sampling horizon T = 256 and SafehlowMatcher for a fixed pre-
diction horizon 7 1 and varying correction horizons 7 E {4, 8, 16, 32, 64, 128, 256) in Maze2D,
when using the closed-form solution of CBF-QP. Each entry reports mean + std over 100 rollouts.
Method (Ciosed-Form CBE)
Score (f)
T-Time
Trap Rate
k())
a
BSI&BS2 (2 0)
RES SifeDif Xiao et al. 2025
1.4/2 = 0. 0.151
1.208
72%
80.30
13 06
398 108.172
1060.86
Yes
SafeFlow Matcher --)
1.610
17%
79:26
262.01
18.19
Safeflow Matcher (7*-8)
1.627 0.018
0.042
0%
75.72
1.64
114.03 6.33
Yes
Matcher (7 161
Matcher
1.634 0.002
0.078
0%
67.96 1.11
89,29 = 1196
Yes
SafeFlow Matcher (3'=
1.634 = 0.003
0.155
0%
67.48
1.09
87.33 0.85
Yes
SafeFlow Matcher (7%-64)
1.633 0.003
0.299
0%
68.03
89.00 0.73
Yes
SafeFloow Mattteeer (7 = 128)
1.632 1: 0.003
0.617
0%
69.72 = 0.98
91.01 = 0.90
Yes
Matcher (7 (7*-256)
1.632 0.003
Table 11: QP-Based CBF Solver: Computation time across correction horizons. Comparison of RES-SafeDiffuser with fixed sampling horizon T 256 and MafcFlowMatcher for fixed predic- tion horizon 7 =1 and varying correction horizons T E [4, 8, 16, 32, 64, 128, 256} in Maze2D, when using the QP solver solution of CBF-QP. Each entry reports mean + std over 100 rollouts.
Method (QP CBF Solver)
Score (t)
T-Time (s)
Trap Rate
k(L)
a(L)
BSI&BS2 (2 0)
SafeDiiiee et 2025
1.488 0.353
9.998
4770.45
2430148
SaachoooMaicher (1"
1,066 0.029
0.157
128
77.31 1. 2.52
276.02
39.01
Yes
SufeFlow (8"==)
1.632 1: 0.0014
0.315
4%
76.93 = 1.11
137.31
1::11:10 11.00
Yes
Safchlooaatcher (T" -16)
1. 1.634 0.003
0.613
07.94 + 1.31
118.72 9.65
SafchlowMatchhr (T"
1.632 0.007
1,247
68.54 1.22
154.20
14.70
Yes
SufeFlowMaccher -64)
1,632 1: 0L0113
2.464
2%
69.54 + 1.44
158.70
23.97
Yes
Malchee (T" 1128)
70.50
174.57
34.67
SuacelowMaccher (16=
1,630 0.004
9.957
13%
67,
1.42
183.47
28.39
Yos
Across both the closed-form and QP-based CBF solvers, SafcFlowMaacher exhibits exceptionally low generation time (T-Time), even when the correction horizon is small. The tables show that Safe- FlowMatcher remains effective and safe much slower and frequently suffers from severe local traps. At T = 4, FaeeFlochtthhe may exhibit minor oscillations near constraint boundaries due to overcorrections caused by the small cor- rection steps (see Figure 14). Although this falls under the definition of the local trap in its impact minimal, in contrast to SafeDiffuser, whose early safety enforcement often leads hard traps and incomplete paths (see Figure 16).
wide range of T values, whereas RES- RES-SafeDiifuser
Figure 14: Local trap of Sathelr T 4. Local traps observed at small correction horizons 7 - 4 in Maze2D. These traps manifest as mild boundary oscillations near obstacles, yet the path remains complete and reaches the goal. Unlike SafeDiffuser depicted in Figure [6] which often fails with incomplete paths under early safety enforcement, SafeFlow maintains path completeness despite minor oscillations.
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F.4 ENERGY- DIIGG---DSTACEEEE ANALYSIS OF DISTRIBUTIONAL DRIFT INDUCED BY CONTROL BARRIER FUNCTIONS
We quantify how much each perturbation Auξ affects the generative process by measuring energy distance between paths with and without safety intervention. For each model pair (Flow Matcher vs. Saacheow,Matcher FM vs. SafeFM, Diffuser (Janerr al., 2022) vs, SafeDiffuser (Xiao al. 2.25 we generate N 100 paths from both the baseline and the corresponding safe variants, starting from the same initial conditions. We define the distance between two paths as the average waypoint-wise wise Euclidean distance
8(T,
E1
H
Given (T Jim and ( L. distance between the two path distributions i8
J where (I.()) denotes the final generated path the (sample) energy
N
De -
base 8(T),
T1.
N
8(T ... Tio)
N
Σ
8( Ti.
safe
i=1
j=1
i=jj=1
i=1j=1
Larger values indicate stronger distributional drift between the baseline and safe path distributions. For each waypoint k, we similarly define a per-waypoint energy distance DE by replacing Ö(T, r') with 8 (T,TT) r') || 7k **1l2 in the above definition of Dg.
SadeFlowMatccerr (ours) SafeFM SafeDiflaser al., 2025)
Euergy Distance per Waypoint 5%
0.7
0.6
0.5
0.4
0.3
0.2
0.1
50
100
200
350
Waypoint Index
Figure 15: Per-waypoint drift between baseline and safe path. For each model pair (Flow- Malcher/SafcFlowWaahhee FM/SafeFM, Diffuser/SafeDifussr), the plot shows the mean per- waypoint deviation between paths produced by the baseline and its corresponding safe variant. The pink band marks the region where the baseline path violates the CBF constraint at the final time; for clarity, single common band is shown, although the exact violation interval may differ by several steps across models.
Figure 15 plots the per- per-waypoint energy distance DP between the baseline and safe paths. Across the three model pairs, the resulting energy distances Di are 0.061 (SafeFlowMatthhr), 0.097 (SafeFM), and 0.229 (SafeDiffuser), showing that SafeFlowMaacher induces the smallest distributional drift while still enforcing safety. The pink band indicates the segment in which the baseline path violates CBF constraint at the final time. Outside this safety-critical region, SafeDiffuser shows large drift, and SafeFM still exhibits noticeable spillover, suggesting that their safety interventions propa- gate to parts the path that do not require correction. In contrast, SafcFlowMMather keeps the drift close to zero outside the pink band.
In SafeFM and SafeDiffuser, the perturbation applied not only at = 1, but also to intermediate generative states T. for E [0, 1) that never executed. Once these perturbed intermediate states
Foor diffusion-hasse samplers, the final path is obtained f rather than f 1, but we use the unified notation Ti,6) for consistency.
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fed back into the velocity field and integrated forward, the resulting deviations accumulate and propagate through the generative dynamics, producing drift at waypoints far outside the final- time violation interval. However, PC integrator in Sawcceeowattcher naturally separate correction from the prediction. It can mitigate this kind of drift effectively.
F.5 VISUALIZATION AND QUALITATIVE ANALYSIS OF LOCAL TRAPS
Following the Definition 2 path is locally trapped if the corrected waypoint exhibits large dis- continuity between two successive corrected path:
|
125.
for some threshold Ç 0. Intuitively, this corresponds to path that get stuck near safety boundaries and consequently produce large jump to escape, often resulting incomplete paths.
SafeDiffuser applies the CBF constraint to each waypoint every sampling step, starting from pure noise. Because the initial waypoints are sampled i.i.d., neighboring waypoints T and 7,- often differ significantly. Since CBFs depend on the waypoint, such large discrepancies cause the resulting CBF corrections to vary greatly across waypoints. Although the diffuser aims to generate contin- uous path (i.e., ||+," -. - S (). applying the CBF constraint independently at each waypoint can break this continuity, pushing different waypoints toward different constraint boundaries and creating local traps. This behavior is clearly visualized in Figures T6 and 17
Mathhlowwatcher, in contrast, begins the correction phase from semi-continuous path TÔ (i.e., ||To T
-ot - S 7) for some small 2 (). Because neighboring waypoints are already close to each other, the resulting CBF corrections vary smoothly across the path. This keeps all waypoints moving in a consistent direction, preserving the path's 'ss continuity and preventing local traps. As visualized Figures and [9] the path maintains forward progress without stalling.
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Figure 16: Path Generation Process of SafeDiffuser (iia et al. 2025 in Maze2D environment with two constraints. From the top-left to the bottom-right, we visualize t, on a uniform time discretization of (T, 0J. excluding the midpoint t = 0.57.
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Figure 17: Path Generation Process of SafeDiffuser (Xiao et al. 2025 in Maze2D environment with six constraints. From the top-left to the bottom-right, we visualize T. on a uniform time dis- cretization of [T, 0). excluding the midpoint t = 0.5T.
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FH
Figure 18: Path Generation Process of Matterr in Maze2D environment with two constraints. Top-left presents the predicted path T" 7 from a noise sample. From the top-left to the bothom-right, we visualize of on uniform time discretization of (0, 1]. excluding the midpoint t t= 0.5.
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FR
Figure 19: Path Generation Process of SafeFlow.Matcher in Maze2D environment with six straints. Top-left presents the predicted path 7. T from a noise sample. From the top-left to the bighhm- we visualize of on uniform time discretization of [0, 1]. excluding the midpoint t 0.5.
30


Reviews:
[{"review_id": "wxZe5w39jq", "reviewer": "Reviewer_Jozp", "summary": "This paper presents SafeFlowMatcher, a novel framework for safe robotic path planning that\neffectively couples generative Flow Matching (FM) with Control Barrier Functions (CBFs). The\nauthors identify a key problem with existing generative planners: they either lack formal safety\nguarantees or, if certification-based, suffer from local trap problems where interventions distort\nthe generative process and cause the plan to fail.", "questions": "● How does the framework handle dynamically changing environments? The CBF-QP is\nsolved at each control step, which seems well-suited for this, but the underlying FM\nmodel is trained to generate paths based on a static map. How would the model perform\nif an obstacle changed position mid-rollout?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "● Paper is very well written and easy to understand\n● The key idea of a two-phase Prediction-Correction (PC) integrator is both simple and\nhighly effective. Decoupling the initial path generation from the subsequent safety\ncorrection elegantly solves the local trap problem\n● The paper provides a formal barrier certificate and a proof of finite-time convergence -\nstrong mathematical guarantees.\n● The method is validated on maze navigation and high-dimensional locomotion tasks,\ndemonstrating superior performance across key metrics: safety, path quality, and\nefficiency", "weaknesses": "● The paper's baselines are exclusively generative models (diffusion- and FM-based) .\nWhile this is the direct field of contribution, classical sampling- or optimization-based\nplanners are still the gold standard in many robotics applications. A comparison against a\nstrong classical baseline (e.g., an optimization-based planner) would have provided a\nbetter picture of the practical performance.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "telqSRh1xG", "reviewer": "Reviewer_ZPuJ", "summary": "The paper proposes SafeFlowMatcher, a planning framework that aims to combine Flow Matching (FM) generative models with the certified safety of Control Barrier Functions (CBFs). The core contribution is a two-phase \"Prediction-Correction\" (PC) integrator. First, a \"Prediction\" phase uses the learned FM model to generate a candidate path in one or a few steps, without safety constraints. Second, a \"Correction\" phase refines this path using a proposed \"vanishing time-scaled flow dynamics\" (VTFD) and a CBF-based quadratic program (QP) to enforce safety and finite-time convergence to the safe set. The authors claim this two-phase approach decouples generation from certification, thereby avoiding the \"local trap\" problem. The method is evaluated on 2D maze navigation and locomotion tasks, showing improved safety and task performance and a zero \"trap rate\" compared to baselines like SafeDiffuser.", "questions": "See the weakness above.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The paper identifies a clear and important problem in safe generative planning. Applying hard safety constraints (like CBFs) directly within the sampling loop of a generative model (like diffusion or flow matching) can distort the learned dynamics, leading to \"distributional drift\" and \"local trap\" failures. The goal of decoupling generation from certification is well-motivated. \n\n2. To further enhance sampling stability in the correction phase, this paper proposes a time-scaled flow dynamics method to contract the prediction errors and mitigate drifting.\n\n3. The paper compares the proposed method with multiple baselines and conducts various ablation studies to demonstrate the efficiency and effectiveness of SafeFlowMatcher.", "weaknesses": "1. The theoretical guarantees for the correction phase (Lemma 2 and 3) rest on strong assumptions about the prediction error $\\epsilon$ (e.g., symmetric, zero-mean, and a locally strongly convex negative log-density). There is no empirical validation to show that the actual integration error from a deep FM model satisfies these convenient properties.\n\n\n\n2. In the experiments (Table 1), SafeFlowMatcher achieves zero trap rate. How does the proposed method guarantee the low trap rate? Can the method consistently achieve a zero trap rate in some higher-dimensional or more complex navigation tasks? Are there any theoretical guarantees?\n\n\n\n3. The current evaluation only includes three simple environments: one maze and two locomotion environments. Can the author provide experimental results on robot manipulation tasks as well? For example, the environment used in SafeDiffuser might be a good starting point.\n\n\n\n4. In Algorithm 1, SafeFlowMatcher first samples a trajectory and solves QP in a double for-loop. The authors report a very fast sampling time of ~4.7ms. To better understand the efficiency of the method, it would be more informative to report the computation time of both phases.\n\n\n\n5. Does the QP always have a solution? What will Algorithm 1 do if the QP can not be solved? Will the safety guarantee still be satisfied in that case? \n\n\n\n6. Does the method always require a specific function $b$ as in equation (6)? How to obtain $b$ for environments with irregular geometry, agents with high-dimensional state space, or robots with image-based observation?\n\n\n\n7. In line 214, \"that places $\\tau_0^p$ sufficiently close to ...\", should the term here be $\\tau_1^p$?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "NuaeQYjsUC", "reviewer": "Reviewer_rRLo", "summary": "The paper introduces SafeFlowMatcher, a planning framework that integrates conditional flow matching with control barrier functions to generate trajectories that are efficient to sample and provably converge to a robust safe set. The method addresses a key limitation of flow-matching and diffusion planners, which produce high-quality paths but lack safety control and can become trapped when constraints are enforced at each latent step.\nSafeFlowMatcher relies on a two-phase prediction-correction scheme. The prediction phase runs standard FM dynamics to propose an unconstrained path, while the correction phase refines it using a vanishing time-scaled FM field and a CBF-based quadratic program that projects waypoints back into the safe set, ensuring finite-time convergence under the derived barrier conditions.\nOn the theoretical side, the authors restate finite-time CBF results and provide a flow-invariance certificate guaranteeing safety for the corrected flow. Empirically, the framework is validated on Maze2D and MuJoCo locomotion. It consistently outperforms FM, DDIM, and SafeDiffuser-based baselines on the Maze2D environment, achieving higher scores while ablations confirm the role of each phase and the vanishing-scale mechanism.", "questions": "- Could you make explicit what you actually assume in the experiments? Am I correct that you do not know the environment transition dynamics and that the only environment-specific information you use is offline trajectories for FM training and analytic, differentiable obstacle/safety functions $b(x)$ with known centers and scales? Is there any case where you used the true simulator for $\\nabla b$? \n- In the maze experiments, are you conditioning FM on start-goal or only on start and letting it flow to the high-density goal region? Could you clarify what is actually fed to the network at inference time?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper identifies a genuine mismatch in existing “safe” generative planners: safety is typically enforced at intermediate latent steps of the denoising or flow-matching process, although only the final trajectory is executed. This “semantic misalignment” is addressed cleanly by decoupling generation and certification: safety is enforced only in the correction phase of the flow. This clarification is timely and relevant for the diffusion/FM-for-planning community.\n- The prediction-correction (PC) scheme is well motivated. The prediction phase runs unconstrained FM dynamics to avoid distributional drift, while the correction phase refines the path using a vanishing time-scaled FM vector field and a one-constraint CBF-QP projecting waypoints into the safe set. The finite-time safety guarantee applies to this corrected flow. The ablation in section 4.2 (prediction-only / correction-only / full PC) clearly supports this design.\n- The QP correction is computationally light: the authors define a convex problem with a single CBF constraint and closed-form projection, countering concerns about runtime overhead. The running time reported in the experiments shows that this approach is quite fast compared to other SOTA methods.\n- The authors performed a serious experimental analysis and comparisons by comparing their approach to a wide variety of baselines (for which they have themselves reproduced the results). Experiments are coherent with the claims and SafeFlowMatcher always achieves a postivie barrier function with zero trap rate, outperforming all baselines while remaining efficient.", "weaknesses": "- On p.3, the authors introduce CBFs in their general control-affine form: $\\dot x = f(x)+g(x)u$, which normally presupposes known dynamics; in experiments, they seem not to use a physical model, but instead plug in analytic obstacle functions. This distinction is not clearly stated.\n- Definition 1 and Lemma 1 are written for general systems with Lipschitz $f,g$, but in practice, the “dynamics” is the FM vector field with vanishing scaling (cf. equation 10). The paper does state this in section 3.2, but the transition is quick. I think the authors should give a short explanation with intuitions; the CBF is not certified in the real environment, but on the generated flow. Furthermore, Definitions 1 and Lemma 1 should be more intuitively justified; currently, they are just \"dropped\" there without a clear explanation, which makes it difficult for the reader to understand their consequences and intuitive meanings.\n- I noticed some clarity issues: on page 3, line 108, some distribution over paths $q$ appears, but has not been introduced anywhere. I also found that, in the main text, the environments and the assumptions made on those environments were not well-specified. For instance, what is the state space and what are the dynamics of the $2$D maze? Does the method assume access to the exact dynamics or just to a dataset containing random trajectories? If this is the case, from where do they come? A short description of the environments and assumptions would help.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pRlJLZfEfM", "reviewer": "Reviewer_gaEu", "summary": "This paper introduces SafeFlowMatcher, a novel planning framework that integrates Flow Matching (FM) with Control Barrier Functions (CBFs) to achieve both real-time efficiency and certified safety in robotic path planning. The core of SafeFlowMatcher is a two-phase Prediction-Correction (PC) integrator. The prediction phase efficiently generates a candidate path using learned FM dynamics without safety interventions. The subsequent correction phase refines this path by (i) reducing integration errors through vanishing time-scaled flow dynamics (VTFD) and (ii) enforcing hard safety constraints using a CBF-based quadratic program (QP), minimally perturbing the vector field. This decoupling strategy prevents distributional drift and mitigates \"local trap\" problems often encountered by other certification-based generative planners. Extensive experiments on Maze2D navigation and high-dimensional locomotion tasks demonstrate that SafeFlowMatcher outperforms existing diffusion- and FM-based baselines in terms of safety (zero trap rate), planning performance (higher scores), path quality (smoother paths), and computational efficiency. Ablation studies confirm the necessity of both prediction and correction phases and the effectiveness of VTFD.", "questions": "See Weaknesses^", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper proposes a unique combination of Flow Matching for efficient path generation and Control Barrier Functions for formal safety guarantees. This addresses a critical need in robotic planning for both speed and reliability. The two-phase PC integrator is a significant contribution. By separating the initial path generation (prediction) from safety enforcement (correction), SafeFlowMatcher avoids issues like distributional drift and local traps that plague other methods that attempt to enforce safety on intermediate latent states.\n2.  The paper provides rigorous mathematical proofs for the forward invariance of a robust safe set and finite-time convergence to this set. This theoretical backing strengthens the credibility of the proposed safety guarantees.\n3. By leveraging Flow Matching, SafeFlowMatcher demonstrates efficiency comparable to or better than diffusion-based methods, which often require many more sampling steps. The ability to achieve high performance with a small number of function evaluations (e.g., $T^p=1$) is a notable advantage.", "weaknesses": "1.  The paper highlights the reliance on well-defined CBFs. Could the authors elaborate on the practical challenges and potential solutions for designing or learning CBFs for more complex, dynamic, or unstructured real-world robotic environments beyond the relatively simple constraints presented? Are there any ongoing efforts to integrate CBF learning within SafeFlowMatcher?\n2. Given that hyperparameter tuning is identified as future work, can the authors provide more insight into the sensitivity of SafeFlowMatcher's performance to the choice of $\\epsilon$, $\\rho$, $\\delta$, $\\alpha$, and $t_w$? Are there any heuristics or adaptive strategies that could be employed to simplify this tuning process for new tasks or environments?\n3. While closed-form QP solutions are mentioned for simple cases, how does the computational overhead of the QP solver scale with a larger number of waypoints ($H$) or a more extensive set of complex safety constraints? Have the authors encountered scenarios where the QP solving time becomes a bottleneck, and what strategies could mitigate this?\n4. The paper mentions developing a guidance-free version as future work. Could the authors expand on the specific challenges in achieving this (e.g., in terms of exploration, multimodal goal reaching) and their initial thoughts on how to approach such a development?\n5. Lemma 1 assumes a symmetric, zero-mean distribution for the prediction error $\\epsilon$. How robust is SafeFlowMatcher to deviations from this assumption, particularly if the prediction errors are biased or exhibit heavy tails? What are the practical implications of such deviations on safety guarantees and performance?\n6. While Naive Safe FM serves as a baseline, could the authors provide a more in-depth discussion on why directly applying safety constraints from the beginning leads to a high trap rate? A clearer theoretical or empirical breakdown of the failure modes of Naive Safe FM would further strengthen the argument for the PC integrator.\n7. For the vanishing time-scaled flow dynamics, the optimal $\\alpha=2$ was found empirically. Is there a deeper theoretical justification for this specific value or for the general range of $\\alpha$ that provides a good trade-off between accelerated convergence and stability?\n8. The current experimental validation focuses on maze navigation and locomotion. How would SafeFlowMatcher perform on more complex robotic manipulation tasks (e.g., pick-and-place with moving obstacles) that typically involve higher-dimensional state and action spaces, more intricate safety constraints, and potentially longer planning horizons?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ypey9PJnpX", "reviewer": "Reviewer_DQsF", "summary": "The paper “Practical Diffusion Planning via Temperature-Guided Reward Conditioning (TGDP)” introduces a diffusion-based planning method that removes the need for task-specific tuning in reward-conditioned decision-making. Building on Classifier-Free Guidance (CFG), the authors propose a temperature-guided framework where trajectory samples are reweighted during training according to a temperature parameter that biases learning toward higher-return trajectories while preserving data diversity. At inference, TGDP employs an adaptive guidance mechanism that dynamically adjusts the influence of reward conditioning based on the geometric alignment of denoising directions across temperature scales, thereby preventing under- or overguidance. Experiments on standard D4RL locomotion, Maze2D, and Kitchen benchmarks show that TGDP achieves performance on par with or exceeding existing diffusion planners, all while using a single default hyperparameter configuration and maintaining computational efficiency.", "questions": "- How sensitive is TGDP to the choice of the **βₘₐₓ** parameter? Could the authors provide guidance or heuristics for selecting it, or explore whether it can be learned automatically?  \n- Since TGDP mainly addresses **guide-scale sensitivity** in CFG, how do the authors envision extending this framework to **other conditioning paradigms** or more general decision-making problems?  \n- The adaptive guidance mechanism is based on **collinearity heuristics** — can the authors provide deeper theoretical justification or empirical evidence (e.g., ablation or visualization) showing that this metric consistently distinguishes inter- and intra-modal cases?  \n- Have the authors evaluated TGDP on **modern or realistic robotic benchmarks**, such as continuous manipulation or navigation tasks with raw sensory input? If not, what challenges do they foresee in transferring TGDP to such settings?  \n- The method adds some computational overhead (~1.5× CFG). Could the authors analyze how this scales with **planning horizon** and **environment complexity**, and whether approximate or truncated versions can retain performance?  \n- Can the **temperature-based reweighting** framework be extended to **learned or adaptive temperature schedules**, potentially improving flexibility across nonstationary datasets?  \n- How does TGDP interact with **online fine-tuning or real-time replanning**, where data distributions evolve — does the adaptive mechanism remain stable in such dynamic conditions?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The paper is well motivated and clearly written, providing a logical flow from problem statement to solution.\n\n- The proposed TGDP method is technically sound and conceptually reasonable, effectively addressing CFG’s sensitivity to guide-scale tuning.\n\n- The method design is elegant and practical, integrating temperature-guided weighting and adaptive guidance without significant computational or architectural overhead.\n\n- Experiments are comprehensive and well executed, covering multiple standard benchmarks (D4RL, Maze2D, Kitchen) and demonstrating consistent performance gains.\n\n- The analysis is insightful, linking theoretical intuition with empirical behavior and validating the method’s robustness and generality.", "weaknesses": "- The novelty is limited, as the contribution primarily targets a known issue (guide-scale sensitivity) within Classifier-Free Guidance rather than introducing a fundamentally new planning paradigm.\n\n- The method introduces additional parameters, such as βₘₐₓ, which slightly increase algorithmic complexity and may require their own tuning for optimal stability.\n\n- The experimental evaluation relies mainly on D4RL, an older and largely saturated benchmark suite; validation on more realistic or modern robotic planning tasks (e.g., navigation or manipulation with sensory inputs) would strengthen the paper’s practical relevance.\n\n- While the adaptive guidance mechanism is elegant, it remains heuristic—its theoretical guarantees or convergence behavior under diverse diffusion architectures are not thoroughly analyzed.\n\n- The temperature-based reweighting is conceptually intuitive but still handcrafted; there is no exploration of whether β or weighting could be learned adaptively or optimized end-to-end.\n\n- The paper’s computational trade-offs (≈1.5× CFG cost) are reasonable but may still pose challenges for real-time or large-scale deployment scenarios.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)