Learning Dynamics under Environmental Constraints via Measurement-Induced Bundle Structures
Authors: Dongzhe Zheng
Wenjie Mei
Venue: ICML 2025 spotlightposter (2025)
Keywords: Learning dynamics
environmental constraints
geometric framework
control barrier functions
Primary area: general_machine_learning->everything_else

Abstract: Learning unknown dynamics under environmental (or external) constraints is fundamental to many fields (e.g., modern robotics), particularly challenging when constraint information is only locally available and uncertain. Existing approaches requiring global constraints or using probabilistic filtering fail to fully exploit the geometric structure inherent in local measurements (by using, e.g., sensors) and constraints. This paper presents a geometric framework unifying measurements, constraints, and dynamics learning through a fiber bundle structure over the state space. This naturally induced geometric structure enables measurement-aware Control Barrier Functions that adapt to local sensing (or measurement) conditions. By integrating Neural ODEs, our framework learns continuous-time dynamics while preserving geometric constraints, with theoretical guarantees of learning convergence and constraint satisfaction dependent on sensing quality. The geometric framework not only enables efficient dynamics learning but also suggests promising directions for integration with reinforcement learning approaches. Extensive simulations demonstrate significant improvements in both learning efficiency and constraint satisfaction over traditional methods, especially under limited and uncertain sensing conditions.

Learning Dynamics under Environmental Constraints via Measurement-Inducee Bundle Structures
Dongzhe Zheng'
Wenjie Mei?
Abstract
detection. Control barrier functions (CBFs) (Ames et al., 2016) have emerged powerful tool for ensuring con- straint satisfaction. However, the classical CBF framework treats measurements as external observations rather than integral components of the system's geometric structure, limiting the ability to fully exploit measurement informa- tion for both constraint verification and dynamics learning.
Learning unknown dynamics under environmen- tal (or external) constraints is fundamental to many fields (c.g., modern robotics), particularly challenging when constraint information is only locally available and uncertain. Existing ap- proaches requiring global constraints or using probabilistic filtering fail to fully exploit the geo- metric structure inherent in local measurements (by using, c.g., sensors) and constraints. This paper presents geometric framework unifying measurements, constraints, and dynamics learn- ing through a fiber bundle structure over the state space. This naturally induced geometric struc- ture enables measurement-awaae Control Barrier Functions that adapt to local sensing (or measure- ment) conditions. By integrating Neural ODEs, our framework learns continuous-time dynam- ics while preserving geometric constraints, with theoretical guarantees of learning convergence and constraint satisfaction dependent on sensing quality. The geometric framework not only en- ables efficient dynamics learning but also sug- gests promising directions for integration with re- inforcement learning approaches. Extensive sim- ulations demonstrate significant improvements in both learning efficiency and constraint satisfac- tion over traditional methods, especially under limited and uncertain sensing conditions.
The fundamental challenge lies in the geometric relation- ship between state space, measurements, and constraint manifolds. Traditional approaches often require complete knowledge of constraints, making them impractical when only local sensors' measurements are available. Consider a robotic arm with force sensors or a drone with range detec- tors even local measurements contain sufficient geometric information about both constraints and underlying dynam- ics, suggesting global knowledge may be unnecessary if we properly exploit this local structure.
Our key insight is that measurement uncertainty naturally induces a fiber bundle structure that unifies measurements, dynamics, and constraints. This geometric perspective re- veals how measurements and dynamics are fundamentally intertwined through the bundle's connection, enabling Neu- ral ODEs (Chen et al., 2018) to learn continuous-time dy- namics that naturally respect the system's 's physical behavior. By leveraging this structure, our approach provides new paradigm for machine learning to understand environmen- dynamics- instead of treating measurements as simple inputs, we exploit their inherent geometric information guide the learning process. The framework allows control strategies to automatically adapt based on measurement quality becoming more conservative in regions of high uncertainty while allowing more aggressive behavior where measurements are reliable.
1. Introduction
Learning unknown dynamics under measurement con- straints fundamental to many applications, from manipu- lators with force sensing to autonomous vehicles with range
The main contributions of this work are: 1) Proposes a novel geometric framework that unifies measurement un- certainty, system dynamics, and constraints within fiber bundle structure, providing principled information for Neu- ral ODEs while maintaining safety guarantees. 2) Introduces adaptive measurement-aware safety certificates (mCBFs, de- fined in Section 3.6) that automatically adjust conservative margins based on local measurement quality. 3) Demon- strates enhanced generalization capabilities across different
Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, China "School of Automation and Key Laboratory of MCCSE of Ministry of Education, South- cast University, Nanjing, China. Correspondence to: Wenjie Mei <wenjie. seu.edu.cn. vmus I 130@@ com...com>.
Proceedings of the nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Learn. Dyn, Environ. Constr. Meas-Innuceed Bundle Struct.
scenarios without requiring global information through ex- perimental validation.
duced Riemannian Motion Policies respecting the underly- ing manifold structure, while (Chen al., 2018) proposed Neural ODEs that opened new possibilities for learning dy- namics with geometric properties. The perspective was pioneered by (Lewis, 1998) establishing connections between mechanical systems and principal bun- dles, with (Montgomery. 1993) developing gauge-theoretie approaches to mechanical control. Building on this founda- tion, (Bronstein et al., 2017; Cohen et al., 2019) advanced geometric methods for learning on manifolds, and (Hansen- Estruch et al., 2021) developed frameworks for control on Lie groups. However, these approaches typically require global geometric information and struggle with local mea- surement uncertainty.
The practical significance lies in learning safely from lo- cal measurements without complete constraint knowledge. By working directly with the bundle structure induced by measurements, we prove that Neural ODEs trained within this framework naturally preserve physical constraints while learning continuous-timg dynamics. This geometric ap- proach provides machine learning algorithms with struc- tured way to understand system dynamics through the lens of measurement geometry, leading to more efficient and interpretable learning. Moreover, this framework offers im- portant insights for reinforcement learning by providing a principled way to handle partial observations and measure- ment uncertainties in the learning process. Experimental results demonstrate the effectiveness in real-world appli- cations, where the learned dynamics model successfully captures both the control-to-trajecton relationships and con- straint requirements under measurement uncertainties. Our implementation is publicly available at https: ub. com Cont nuumCoder /Messuremettt Induc ed- Bundle- for /
2.3. Measurement-Aware Control
The geometric treatment of measurement uncertainty draws inspiration from (Hsu, 2002) on stochastic differential ge- ometry and (Diaconis et al., 1988) on geometric filtering theory. Early approaches like (Kalman, 1960) developed robust control using filtering- -ased state estimation, while (Berkenkamp et al., 2017) proposed learning control un- der measurement noise. Recent advances by (Wu et al., 2015) introduced geometric numerical integration methods for uncertainty quantification, while (Boumal, 2023) devel- oped comprehensive tools for optimization and estimation on manifolds. However, these works often treat measure- ment uncertainty as an external disturbance rather than an intrinsic geometric property of the system. Our framework addresses these limitations by unifying measurements, con- straints, and learning objectives through the natural fiber bundle structure induced by the measurement process, en- abling more efficient learning while maintaining rigorous safety guarantees.
2. Related Work
2.1. Safety-Critical Control and Learning
Our work builds upon funndaeental theoretical advances in differential geometry and control theory. The fiber bundle framework we employ originates from Ehresmann's 's sem- inal work (Ehresmann, 1950) on geometric connections, later developed by (Kobayashi & Nomizu, 1996) for con- trol applications. Early work in safety-critices control fo- cused on analytical safety certificates through Control Bar- rier Functions (CBFs). (Ames et al., 2019) introduced the foundational CBF framework providing formal guarantees for constraint satisfaction in known dynamical systems, ex- tended by (Jankovic, 2018; Daş & Murray, 2022; Choi et al., 2021) to handle bounded disturbances through robust CBFs. Learning-based approaches emerged address model un- certainty while maintaining safety guarantees. (Cheng et al., 2019) proposed Neural CBFs that learn safety certificates directly from data, while (Taylor et al., 2020) developed the LafeLearn framework combining Gaussian processes with CBFs for safe exploration. However, these methods treat measurements as perfect observations rather than uncertain quantities, limiting their real-world applicability.
3. Theoretical Foundations and System Modeling
Through fiber bundle structures and measurement-adapted barrier functions, this work establishes a geometric frame- work that unifies the challenge of maintaining safety guaran- tees while adapting to uncertainties in both system dynamics and measurements in safe learning control.
3.1. System Model and Measurement Structure
Let T.M denote the tangent space at point x E M,, which is the vector space of all tangent vectors at æ, and TM .MM denote the cotangent space at x and T M = U T M represents the cotangent bundle.
2.2. Geometric Learning and Bundle Theory
UzeM T be the tangent bundle. Let T:M
Geometric structure preservation in learning control has seen significant development, building on (Marsden & We- instein, 1974)'s theoretical foundation for geometric me- chanics and symmetry reduction. (Ratliff et al., 2018) intro-
Consider controlled dynamical system with state M on smooth manifold M and control input u E 2 C
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Learn, Dyn, Environ. Constr, Meas--Induced Bundle Struct.
RR The system evolution and measurement process are described by
3.4. Uncertainty Propagation
The propagation of uncertainties through the bundle struc- ture follows from the differential geometry of the fiber bun- dle. The key relationships are:
④ = ==(xx, + ((z)w, =00)= = D0
(1)
y = //x)+(())))))))))) +
where f: .M U + TM represents the nominal dynam- ics, 9: .M- - T characterizes model uncertainty, and h: M + 3/ is the measurement map. The process noise w and sub-Gaussian measurement noise are bounded with |w] < 8n and |o| < 8,, respectively.
dx,(X,) = [(x,
du, (Xg) = g(z)w
(5)
dy dd(xx) + de
where dı, represents the pushforward of the vector fields x,, X, along the projection æ. These relationships induce an uncertainty tube Te t) around nominal trajectories:
The measurement space y S I carries a natural metric structure induced by the Euclidean norm: (2)
v ( 32.1)² This metric quantifies the uncer- tainty measurement space and plays a crucial role in safety analysis.
Te(x, {y dy(u. h()x()))) S <()))
(6)
where Φ2 denotes the flow of the nominal system and e(t)
3.2. Fiber Bundle Framework
characterizes the growth uncertainty over time.
The relationship between states and their measurements induces a natural fiber bundle structure :: & + -M where MM x y is the total space. For each state M, the fiber π 1 (3:) characterizes the set of possible measurements:
3.5. Compatible Group Actions
The system often exhibits symmetries that can be exploited for, for example, dimensional reduction. These symmetries are captured by compatible Lie group actions. Let G be Lie group acting on both M and y through smooth maps. Then,
π (x) = (xx y) E E = h(x) + v, ||||| S 5.} (2)
This bundle is equipped with a connection that describes the geometric relationship between system trajectories and measurement evolution:
M G M
y G x y
X = -. -..'
(3)
The compatibility conditions for these actions are
where Vx Y represents the covariant derivative of the vector field Y (EE) along the vector field X ((E) ((E) denotes the space of smooth vector field on 8), m. refers to the pushforward map of the projection æ, K: .M L()), TM) is the measurement feedback gain operator that couples state and measurement dynamics. Here, L(), TM) denotes the space of bounded linear operators from y to T.M.
f(WM(g.2) u) di (g, (z) N(V x)) (g M(x))
(8)
for all g E G. where di (g. represents the differential of the map V (M with respect to the state. These conditions ensure that the symmetries respect both the dynamics and measurements.
3.3. Bundle-Based Safety Certificates
3.6. Measuremenn--Aapted Control Barrier Functions
Safety constraints are formalized through smooth bundle map 4: & - over the fiber bundle ): E - M satisfying three fundamental properties:
The cornerstone of our safety framework is the concept of measurement-addppeed Control Barrier Functions (mCBFs). A smooth function b: E + R qualifies as an mCBF if it satisfies
1. (xx, h(x)) > 0 for all E Sp
2. The bundle derivative dd(() > 0 for all X E A(E),
1. b(x y) 2 0 = x E So
where d (X): = (, X)
2. inf well
L + (L, + a(b) 2
(9)
4 (xx v) 2 <(yy lI) for some 7 E Koo
(4) Here, So denotes the nominal safe set, A(E) is the space of admissible vectors on the total space E, V 8 is the covariant derivative on E, and y is a class Ko function that captures the degradation of safety guarantces with measurement un- certainty.
[b(,, yı) b(æ, y2)1 S Lody (111 82)
where Ly and Lg denote the Lie derivatives along vector fields ff and g respectively, a is a class Ko function, and L& > 0 is the Lipschitz constant of with respect to mea- surements (recall that Sn denotes the nominal safe set).
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Learn, Dyn, Environ. Constr, Meas--Inuueed Bundle Struct.
3.7. Safety Guarantees
where C (E) denotes the space of smooth functions de- fined on the total space &, ((TE) stands for the space of sections of the tangent bundle Tε, (x,) y) represents the operator L acting on Q, evaluated at the point (z, v), VE is the connection defined on E, and R provides a reg- ularization function preserving the fiber structure of the bundle.
The culmination of this geometric framework is captured in the following fundamental theorem:
Theorem 3.1. Given an mCBF satisfying the preceding conditions, if ((x()), y(0)) 2 0, then for any admissible noise sequences w() v()):
P(x(t)= E for all t 2 20) 21 cx-cc(((c6
(10)
The learning dynamics on the bundle take the form
where e > is a constant depending on system parameters.
-c -) f) $ C2 - (8)
(12)
Proof Sketch. Sketch. The proof proceeds through three key steps. First, establish that the bundle connection preserves safety certificates along fibers, utilizing the compatibility conditions between the connection and barrier function. Sec- ond, we demonstrate that uncertainty propagation remains bounded within the tube Te, leveraging the Lipschitz prop- erties of the system dynamics. Finally, we show that the Lipschitz condition on b ensures controlled variation of safety certificates under measurement uncertainty, leading to the probabilistic bound (1 exp( exp(=c/ō.)). The technical proof is provided in Appendix A.
where Li, L2 are compatible bundle-valued operators. f denotes the learned estimate of the true system dynamics f. while 4' represents the optimal barrier function that ensures safety guarantees, both serving as target values in the learning dynamics governed by bundle-valued operators.
4.2. Measurement-AAapted Safety Certificates
Let b E + IR denote the nominal safety certificate that characterizes system safety under ideal measurements. The safety certificate x: & R adapts measurement uncer- tainty through:
The geometric framework developed in this section estab- lishes three key theoretical advances. First, the fiber bundle structure provides a natural setting for handling measure- ment uncertainty, enabling precise tracking of error prop- agation through system dynamics. Second, the compati- ble group actions facilitate systematic dimension reduction while preserving safety properties through quotient space dynamics. Third, the measurement-adappe adapted Control Bar- rier Functions yield robust safety guarantees that degrade gracefully with measurement noise, thanks to their Lipschitz continuity properties.
(xr = o0(r, allyy - h(=))l)
(13)
I + (Igg 2 (3) along solutions
where $ the nominal certificate, a E Koo. and B is a class K. function.
4.3. Uncertainty-Aware Learning Algorithm
The learning process incorporates measurement uncertainty through:
This theoretical foundation directly enables practical leamn ing algorithms that maintain safety under realistic sensing conditions, as we will demonstrate in the subsequent sec- tion.
Ô -=-AVoT
N T(f,D) = Σ 1f((,, uz) -bal?
(14)
where Σ, captures measurement uncertainty in data point i, The learning rate matrix A guides parameter updates, while I. 1 denotes the uncertainty-weigheed norm using the inverse covariance matrix Σ, and D contains N triplets of state, input, and state derivative measurements.
4. Learning Framework under Measurement Uncertainties
Building on the geometric foundations, we now develop a learning framework that actively incorporates measurement uncertainty. The key idea is to learn both the dynamics and safety certificates the bundle E.
4.4. Santty-Constrained Policy Updates
Let θ E Rd denote the parameters of a policy Te : M U that maps state to control inputs. The policy update law preserves safety through:
4.1. Valued Learning Operators
Define the bundle-valued learning operator L: (8) I(TE):
é [Is(-V)J(OO)
(15)
(x,) (
(11)
S (O: : : v) 2 0 for all (x. v) E

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Learn. Dyn. Environ. Constr. Meas--nnuceed Bundle Struct.
where g denotes projection onto the safe policy set S, and ((O)= Ero Di= 7 ( (z)))) represents the
and system-level guards, though they still lack explicit han- dling of measurement uncertainty. Physics-Informed and Geometric Methods: To evaluate our physical consistency, we compare against PNDS (Djeumou et al., 2022) and GEM (Hansen-Fistruch et al., 2021), which encode physical laws through specialized neural architectures. We also in- clude GeoPath (Zhang et al., 2015) that leverages geometric principles for control design, though without addressing measurement uncertainty. Robust and Adaptive Control: Several approaches address system robustness through dif- ferent theoretical frameworks. RobustSafe (Gurriet et al., 2020) provides safety-critical control using fixed uncertainty bounds, while DataFilter (Wabersich et al., 2023) leverages data-driven safety filters for handling uncertainties, and AdaptSafe (Taylor & Ames. 2020) introduces measurement- dependent barrier functions. Uncertainty-Awant Predic- tive Control: For handling uncertain dynamies, GPMPC (Bonzanini et al., 2021) and ALMPC (Saviolo et al., 2023) employ Gaussian processes and active learning for uncer- tainty quantification. SafeRL (Cheng et al., 2019) com- bines reinforcement learning with barrier functions, demon- strating strong performance in handling model uncertainty through probabilistic frameworks, though these methods lack formal geometric safety certificates.
pected discounted cumulative cost under policy θ with immediate cost c(x, (z)),, discount factor 7 E (0, 1), and initial state distribution z0--
4.5. Convergence and Safety Guarantees
Building on Theorem 3.1, we establish the convergence properties of our learning framework:
Theorem 4.1. Under the proposed learning dynamics, we have
Ilf - flle= S expp(-211) + c200
(16)
((x((1) E So) 2 1- - exp(-ca/85
where c1 C2 C3, A1 > are constants.
A detailed proof of Theorem 4.1 is in Appendix B. The first inequality shows exponential convergence of the learned dynamies with a residual error bounded by measurement uncertainty, while the second preserves the safety guarantees during learning.
5. Experimental Design
We design a comprehensive experimental framework to evaluate our proposed method against state-of-the-art ap- proaches, focusing on three interconnected research direc- tions: Learning-based safety control, geometric structure learning, and safe control under uncertainty. The experi- ments are constructed to highlight key methodological differ- ences while ensuring fair comparison through standardized implementations and evaluation protocols.
5.2. Experimental Tasks
We implement three tasks simulation environment built Genesis physies engine (Xian al., 2023). The first task examines a soft-body worm robot (0. 1m per segment) navigating through obstacles to reach target, using a fixed- step forward Euler integrator (dt = 5e-4s). The worm is modeled using the Material Point Method (MPM) with the neo-Hooken material model. Let E R denote the posi- tion field and p the material density, the dynamics follow:
5.1. Baseline Methods
For detailed discussions on comparisons with mainstream advanced manifold learning methods, we refer readers to Appendix F. While direct numerical comparisons with re- cent geometric deep learning approaches may seem natural, there are several fundamental differences that make such direct benchmarking potentially misleading. We evaluate our method against -fte-offthheart approaches spanning different technical directions in safe learning control. Our baseline selection aims to comprehensively compare with methods that address various aspects of our proposed frame- work:
X = PPPb
(17)
where P is the first Piola- Mirc-- stress tensor and b represents body forces. For neo- neo-Hookean materials:
P= + Alog(J)F
(18)
Here F is the deformation gradient, J = det(F), and material parameters μ. X are unknown. The control input consists of four muscle actuation signals u = [un,, Uhr, Wif, unn] E [o, 1]s, where subscripts indicate upper-fore. upper-hind, lower-fore, and lower-hind muscle groups respectively. The system must maintain safe dis- tances from obstacles through constraints h;(x) = ||x Xohx,ill2 Te 2 0 for all obstacles = 1,..., No Six visual sensors (two on head/tail, four on sides) provide local measurements y, = [x [x-33 -Xhhe ||x - Xab=lla|T + v, with un- certainty bound ||v,]| I <a||x- - x,|| increasing with distance from sensor location x;.
Learning- based Safety Certification: We implement Neural-CB. (Liu et al., 2023), BayesSafe (Berkenkamp et al., 2023), and StructCBF (Taylor et al., 2020) as fun- damental approaches using neural networks and Bayesian optimization for safety certification. Recent advances like SafetyNet (Vitelli et al., 2022) and SafeTrack (Li et al., 2024) enhance these guarantees through adaptive barriers
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Learn, Dyn, Environ. Constr. Meas--Inuueed Bundle Struct.
The second task involves 7-DOF Franka arm performing obstacle-aware manipulation, integrated with dt le-2s. Let IR? denote the joint angles and (q) the inertia matrix, the system dynamics with control input T follow:
states are sampled ensure feasible paths exist while pro- viding sufficient challenge for evaluating the learning and control performance.
5.3. Implementation Details
M(q]q + C(q q) + g(q) + f(4) = (19)
All experiments are implemented in Python using Py Torch, with Soft Actor-Critic (SAC) learning framework. For fair comparisons, we maintain the original implementations for model-based baselines (GPMPC, RobustSafe, ALMPC) and learning-baa basedd base- lines (Neural-CBF, SafetyNet, DataFilter). Our neural ar- chitectures use three hidden layers (128-64-3 188664322 units) with ReL.U activations, while barrier functions add tanh activa- tion in the output layer for boundedness. All networks are trained with Adam optimizer using mixed precision train- ing on an NVIDIA RTX 3090 GPU. Detailed implemen- tation specifications, including hyperparameters, network architectures, and optimization techniques, are provided in Appendix C.
where C(q. q) represents Coriolis terms, g(q) gravity, f(q) joint friction, and T control torques. The system operates der joint limits (q) 9mar qq 2 0 and obstacle avoid- ance constraints (q) = (q) Possll2 fa 2 O, where Pee (q) and Pobs denote the end-effector and obstacle positions, respectively.
base reinforcement
The third task features a quadrotor drone navigating through 3D space, integrated with dt = 2e-3(sec). Let p E R2 denote position, v E R velocity, R E SO(3) orientation, and w E R angular velocity, the dynamics are described by
P
Rf
g0g D(v) R|w|x
m
(20)
For the soft worm task, uncertainties include MPM material parameters (u, À variations of + 10%), actuation response (#:5% muscle force scaling), and sensor noise proportional to distance (0.5-2% of measured distance). The Franka arm experiments incorporate joint friction variations (+88),, payload changes (0-200g), and measurement uncertainties in joint angles (=0.02 rad) and end-effector pose (#2cm). The quadrotor tests feature mass variations (=5%), aerody- namic disturbances (up to O.2N), and depth measurement noise scaling with distance (1-3% of measured depth).
R
(T Jw)
where m is mass, unknown gravity, D(v) unknown aero- dynamic drag, J inertia matrix, and [ the kew-syymmetric matrix operator. The control input is generated through four rotor speeds wi, with unknown thrust coefficient ky- The quadrotor must maintain safe distances from obstacles through constraints (p(p) ||p Pobs,ill2 dsafe O. Four visual sensors provide depth measurements y/ ||p - Pabs,i ||2 + ny in front, back, left and right directions, with uncertainty bound lei S 7/Ip- - Poba,ill2 proportional to depth. Here a, 8. y are unknown uncertainty scaling factors.
5.4. Evaluation Metrics and Protocol
We evaluate both motion quality and safety performance using comprehensive metrics including success rate (SR), path efficiency measures, safety margins, and control quality indicators. Detailed definitions and calculations of these metrics are provided in Appendix C.I. All experiments are conducted in 2m 2m x 2m workspace with randomly generated start/goal positions and obstacle placements. We test the worm robot (500 trials), Franka arm (400 trials), and quadrotor (300 trials) under various task scenarios. Com- plete experimental settings and success criteria are detailed in Appendix C.2.
Figure 1. Illustration of three experimental tasks from left to right: coft-bodied worm robot navigating through obstacles using peri- staltic motion, 7-DOF Franka robotic arm performing obstacle- aware joint motion, and quadrotor drone executing 3D navigation. Blue spheres indicate initial positions, yellow arrows represent motion trajectories, green spheres mark target positions, and red objects denote obstacles.
6. Results and Analysis
We evaluate our method across three robotic control tasks demonstrate generalization of safety constraints under novel obstacle configurations with local observations. Table I presents the quantitative results.
For all three tasks, the workspace is configured as 2m x 2m x 2m arena with randomly placed obstacles. The obstacles' positions are sampled uniformly within the workspace while maintaining minimum separation distances. Initial and goal
While learning-based safety approaches achieve limited success rates (82%-86%) with fixed barrier functions that cannot adapt new configurations, and physics-informed
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methods maintain geometric properties but achieve only 73%-766 success in dynamic environments, our method's measurement-induuee bundle structure enables 96.3% suc- cess rate. Traditional uncertainty-awe MPC methods achieve high constraint satisfaction (99.7%) but produce overly conservative trajectories (26-27m vs our 18.5m), lack- ing our geometric framework for measurement uncertainty.
7. Ablation Study
To validate our design choices and understand the inter- play between different components, we conduct compre- hensive ablation studies focusing on three key architectural elements: (1) mecsuremenntinnuucee bundle structure, (2) measurement-aware CBFs (mCBFs), and (3) Lie group sym- metry.
The key advantage of our approach lies in the unified ge- ometric treatment of measurement uncertainty and safety constraints. Unlike recent adaptive methods that handle un- certainty estimation and safety certification separately (88%- 89% success), our framework enables simultaneous adap- tation of safety bounds and uncertainty estimation through the fiber bundle structure. This fundamental integration of measurement uncertainty into the geometric safety con- straints allows our method to achieve superior performance (96.3% success, 18.5m paths with 99.3% constraint satis- faction) while maintaining robust safety guarantees across novel environments,
For each ablation variant, we perform 10 independent runs on each of the three experimental tasks (soft robot naviga- tion, Franka arm manipulation, and quadrotor control). The success rate (SR) represents the percentage of successful task completions averaged across all runs and tasks. Table 2 presents the comparative results.
Further ablation experiments examining the impact of differ- ent measurement uncertainty patterns (c.g.. Gaussian noise, bias, delays) and bundle structure variations (e.g.. fixed vs. adaptive dimension) are presented in Appendix E.
The measuremenntinduued bundle structure provides geo- metric framework for handling system uncertainties, leading to more efficient trajectories and improved success rates, Without this structure, the success rate drops significantly from 96.3% to 62.7%, while path length and control magni- tude increase substantially, indicating degraded performance and efficiency.
6.1. Performance Convergence Analysis
Figure 2 showcases the training convergence trends of our proposed method in comparison with selected baseline ap- proaches across three distinct tasks: Soft Worm Peristaltic Navigation, Franka Arm Joint Motion, and Quadrotor Pro- peller Control. Across all tasks, our method demonstrates significantly faster convergence rate, reaching optimal per- formance metrics within fewer training episodes than the baseline methods. Additionally, the shaded regions repre- senting standard deviation are noticeably narrower for our method, indicating reduced performance variance and en- hanced stability during training. In the Soft Worm Peristaltic Navigation task, our approach achieves higher average re- turns more swiftly, highlighting its efficiency in simpler navigation scenarios. For the more complex Franka Arm Joint Motion and Quadrotor Propeller Control tasks, our method not only converges rapidly but also maintains higher final performance levels with minimal fluctuations, under- scoring its robustness and reliability in handling intricate control dynamics.
Removing mCBFs reduces our framework to its underlying Neural ODE architecture, which focuses solely on learn- ing dynamics without safety constraints. This leads to the most severe performance degradation, with the success rate dropping to 45.7%. The dramatic reduction in constraint satisfaction rate (from 99.3% to 72.8%) demonstrates that while Neural ODEs can effectively learn system dynam- ies, they struggle to maintain safety constraints without the geometric safety certificates provided by mCBFs. The substantially increased path length (48.2m vs 18.5m) and control magnitude (0.85 vs 0.17) further suggest that pure dynamics learning leads to inefficient and potentially unsafe trajectories.
The Lie group symmetry enables dimension reduction and invariant control synthesis, which is particularly beneficial in tasks involving rotational and translational symmetries. Its removal shows relatively mild performance degradation (success rate of 85.7%), suggesting its role as a complemen- tary enhancement to the core geometric framework rather than a critical component.
6.2. Noise Robustness Performance Experiment
Our approach demonstrates remarkable stability across all noise levels 10-33) for all three tasks. The averaged success rates stay above 91% with minimal variance, while baseline methods like Neural-CBF and SafetyNet show sig- nificant degradation under higher noise conditions (Fig- ure 3). This robust performance demonstrates how the measurement-induced bundle structure naturally handles uncertainties in different scenarios, validating that our ge- ometric safety certificates effectively preserve constraints regardless of the underlying task complexity.
8. Conclusion
This paper presents novel geometric framework that unifies measurements, constraints, and dynamics learning through a fiber bundle structure. Our framework provides funda- mental insights into how dynamical systems can learn and
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Learn, Dyn, Environ. Constr. Meas--Inuuced Bundle Struct.
Method
(%)
(m)
FSE
MMC (m)
AMC (m)
CSR (%)
ACM
Ours
96.3
18.5
383+ 18
0.05=0.01
0.24+00.03
0.26=0.03
99.3
0.17:00.02
0.03++0.004
Neural-CBF
84.0
22.3:10.8
425::19
0.12:0002
0.231:004
0.22-0.03
98.7
0.25:10.33
0.02:10.005
SafeLearn
82.3
23.8+0.9
428:121
0.1320.02
0.221:004
0.21=0.03
98.3
0.2611003
0.04:10.006
SafetyNet
86.7
22.1++0.7
420:1:18
0.11+0002
0.25-+0.03
0.23+0002
98.7
0.24:10.02
0.03-+-0.005
SafeTrack
85.3
22.4:10.8
423:1:9
0.12:10.02
0.24:10.03
0.22:10.03
98.7
0.25:10.03
0.03:10.005
StruciCBF
75.7
2..111.0
435:1:22
0.15
0.27:10.03
0.25
97.7
0.28:10.04
0.05:10.007
PNDS
73.3
1.
438+223
0.16##0.3
0.26++0.04
0.24=0.03
97.3
0.29++0.04
0.05++0.008
GeoPath
74.0
24.2+1.0
436++22
0.15+00.3
0.26++0.03
0.24+0.03
97.7
0.28++0.04
0.05++0.007
GEM
76.3
24.0+0.9
433+-21
0.14+0.02
0.25++0.03
0.23+0.03
98.0
0.27++0.03
0.04++0.006
GPMPC
67.7
26.9=0.9
452:4:00
0.18=0.02
0.23+40.04
0.22=0.03
99.7
0.26+4003
0.04:10.006
ALMPC
71.7
26.2.10.7
442:118
0.161:0.02
0.25:10.03
0.23:10.02
99.0
0.24:10.02
0.03:10.005
SufeRL
70.3
26.410..8
444::19
0.17:10.02
0.24:10.03
0.22:10.02
98.7
0.25:10.03
0.04:10.006
RobustSafe
70.0
26.5++0.8
446+119
0.17+0.02
0.24++0.03
0,23+0002
98.7
0.25++0.03
0.03++0.005
AdaptSafe
88.3
21.8+0.6
417+17
0.10+0.01
0.25+
0,24+0002
99.3
0.22++0.02
0.03++0.004
DataFilter 89.7
21.7:40.6
415+-16
0.10
0.26+00.03
0.24+0002
99.3
0.20++0.02
0.02++0.006
Table 1. Average performance comparison across soft robot navigation, Franka manipulation, and quadrotor propeller control tasks.
SR: Success Rate, PL: Path Length, GRS: Steps to Complete, FSE: Final State Error, MMC: Minimum Safety Distance, AMC: Average Safety Margin. CSR: Constraint Satisfaction Rate, ACM: Average Control Magnitude, CS: Control Smoothness.
Sut Warm Peristafio Navigation
Franki Arm Juin Motion
Quadrotor Propelior Contral
5.5
Ous
D2
Neund
StrecCOF
0.0
GPMING
06
Nourll CDF
Sava
04
StratCos
04
aPssnnc
02
DataFibbr
Epicodes
Eploodos
Epioddos
Figure 2. Training convergence trends. From left to right are soft worm navigation, Franka robotic arm manipulation, and quadrotor control tasks. Each subplot shows the average return and standard deviation range across 10 trials.
Success Rate under Differant Noise Levels
Metries
Full Model
w/o Bundle
w/o mCBF
w/o Lie Group
100
SR
96.3
62.7
45.7
PL (m)
18.510.77
35.11139
48.2.:5.1
22.51:1.5
90
383+-18
712-882
935-108
FSE
0.0510.01
0.15::0:03
0.21
0,0810.02
80-
MMC (m)
0.24:00.03
0.18:10.04
0.15=0005
0.2240.03
75.5
AMC
0.26:00.03
0.20=:0:03
0.17=0.04
0,24+00.22
CSR (%)
99.3:10.2
88.2:2244
72.823.6
96.7+1.0
Success
R0D
ACM
6.17+0.02
0.51++0.12
0.85-0.18
0.26+00.55
CS
0.03:20.004
0.07=0.008
0.09+00010
0.04=0.005
Table 2. Ablation study results across three experimental tasks.
(0-0.1)
(0+0.2)
(d-0.33)
Oura
CPSPC
DMiihiber
teract with their environment through local sensing, while the measurement-aware Control Barrier Functions enable adaptive safety certificates that emerge from direct environ- mental interactions rather than prescribed global knowledge.
Figure 3. Average success rates across three tasks under differ- ent noise levels. Comparison between our method, Neural-CBB SafetyNet, GPMPC and DataFilter. Error bars indicate standard deviation across 10 runs.
Limitations and Future Work Despite these advances, our current implementation has limitations in handling highly stochastic dynamics and complex environmental un- certainties. Future work could explore richer representations of environment-agenn interactions and investigate more so- phisticated uncertainty quantification methods for embodied learning. Additionally, the framework could be extended to better understand how local observations can build towards
adapt under environmental constraints through local obser- vations, bridging the gap between theoretical control guar- antees and modern robotics (or even practical embodied intelligence). The ceesuurmmen-indducee bundle structure naturalyy captures how autonomous agents perceive and in-
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global understanding of environmental constraints and dy- namics, potentially offering new perspectives on embodied intelligence and adaptive control. These results establish promising direction for understanding physical systems learning through environmental interaction, offering insights into, for example, both dynamical systems theory and em-
autonomous systems, ultimately benefiting society through improved safety, efficiency, and technological accessibility. The framework's emphasis on measurement-awwre safety could serve as foundation for future developments in safe autonomy, potentially influencing standards and best prac- tices in the field.
bodied intelligence principles.
References
Acknowledgments
Ames, A. D., Xu, X., Grizzle, J. W., and Tabuada, P. Control barrier function based quadratic programs for safety crit- ical systems. IEEE Transactions on Automatic Control, 62(8):3811-3876 2016.
This work was supported in part by the National Natural Sci- ence Foundation of China (NSFC) under Grant 62403125, in part by the Natural Science Foundation of Jiangsu Province under Grant BK20241 1833, and part by the Fundamental Research Funds for the Central Universities under Grant 2242024k30037 and Grant 2242024k30038.
Ames, A. D., Coogan, S., Egerstedt, M., Notomista, G., Sreenath, K., and Tabuada, P. Control barrier functions: Theory and applications. In 2019 18th European Control Conference (ECC), pp. 3420-3431. IEEF, 2019.
Impact Statement
Berkenkamp, F., Turchetta, M., Schoellig, A., and Krause, A. Safe model- reinforcement learning with stability guarantees. Advances in Neural Information Processing Systems, 30, 2017.
Our work advances both theoretical understanding and prac- tical capabilities in safe learning control systems, with sig- nificant potential impacts across multiple domains. The framework developed in this paper enables more reliable and safer operation of robotic systems in uncertain envi- ronments, which could substantially reduce accidents and improve human-robot interaction safety in manufacturing, healthcare, and service robotics. This enhanced safety mech- anism is particularly valuable as autonomous systems be- come more prevalent daily life.
Berkenkamp, F., Krause, A., and Schoellig, A. P. Bayesian optimization with safety constraints: safe and automatic parameter tuning in robotics. Machine Learning, 1 112(10): 3713-3747, 2023.
Bonzanini, A. D., Paulson, J. A., Makrygiorgos, G., and Mesbah, A. Fast approximate learning- based multistage nonlinear model predictive control using gaussian pro- cesses and deep neural networks. Computers Chemical Engineering, 145:107174, 2021.
The geometric approach introduced here provides new theo- retical insights into the relationship between measurement uncertainty and safety guarantees, contributing to the fun- damental understanding of safe learning systems. Further- more, by better handling measurement uncertainties, our approach can reduce the need for expensive sigeppreciioon sensors, potentially making advanced robotic systems more accessible and cost-effective. This democratization of tech- nology could accelerate innovation and adoption of safe autonomous systems across various industries.
Boumal, N. An introduction to optimization on smooth manifolds. Cambridge University Press, 2023.
Bronstein, M. M., Bruna, J., LeCun, Y., Szlam, A., and Van- dergheynst, P. Geometric deep learning: going beyond euclidean data. IEEE Signal Processing Magazine, 34(4): 18 02-42,20 2017.
While any advancement in autonomous systems technology warrants careful consideration of its deployment context, our framework is designed with robust safety guarantees that help ensure responsible implementation. The theoretical foundations and practical demonstrations provided in this work establish clear guidelines for appropriate use cases and system limitations. We are committed to open-source release of our implementation to promote transparency and reproducibility, maintaining comprehensive documentation about system capabilities and intended applications.
Chen, R. T., Rubanova, Y.. Bettencourt, J., and Duvenaud, D. K. Neural ordinary differential equations. Advances in Neural Information Processing Systems. 31, 2018.
Cheng, R., Orosz, G., Murray, R. M., and Burdick. J. W. End-to-end safe reinforcement learning through barrier functions for safety-critical continuous control tasks. In Proceedings of the AAAI Conference on Artificial Intelli- gence, volume 33, pp. 3387-3395, 2019.
Choi, J. J., Lee, D., Sreenath, K., Tomlin, C. J., and Her- bert, S. L. Robust control barrier-value functions for safety-critical control. In 2021 60th IEEE Conference on Decision and Control (CDC), pp. 6814-6818 6821- IEEE, 2021.
Through thoughtful deployment and continued refinement of safety mechanisms, we believe this work will contribute positively the development of more reliable and accessible

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Learn, Dyn, Environ. Constr. Meas--Induced Bundle Struct.
Mathematical Physics, 42(1):135-164, 1998. Proceed- Cohen, T., Weiler, M., Kicanaoglu, B., and Welling, M. Gauge equivariant convolutional networks and the icosa- hedral cnn. In International Conference Machine Learning, Pp. 1321-12211 11330. PMLR, 2019.
ings of the Pacific Institute of Mathematical Sciences Workshop Nonholonomic Constraints in Dynamics.
Li, X., Li, Y., Girard, A., and Kolmanovsky, I. System-level safety guard: Safe tracking control through uncertain neu- ral network dynamics models. In 6th Annual Learning for Dynamics & Control Conference, pp. 968-979. PMLR, 2024.
Daş, E. and Murray, M. Robust safe control synthesis with disturbance observer-based control barrier functions. In 2022 IEEE 6l Conference Decision and Control (CDC), pp. 5566-5573, IEEE, 2022.
Diaconis, P., Elworthy, D., Föllmer, H., Nelson, E., Papani- colaou, G., Varadhan, S. R., and Elworthy, D. Geometric aspects of diffusions on manifolds. In École d' d'Ééé de Probabilités de Saint-Flour XV-XVII, 1985-87, 87, pp. 277- 425. Springer, 1988.
Liu, S., Liu, C., and Dolan, J. Safe control under input limits with neural control barrier functions. In Conference on Robot Learning, pp. 1970-1980. PMLR, 2023.
Marsden, J. and Weinstein, A. Reduction of symplectic man- ifolds with symmetry. Reports on Mathematical Physics, 5(1):121-130, 1974.
Djeumou, F., Neary, C., Goubault, E., Putot, S., and Topcu, U. Neural networks with physics-informed architec- tures and constraints for dynamical systems modeling. In Learning for Dynamics and Control Conference, pp. 263-277. PMLR, 2022.
Montgomery, R. Gauge theory of the falling cat. Fields Institute Communications, 1(10.1090), 1993.
Ratliff, N. D., Issac, J., Kappler, D., Birchfield, S., and Fox, D. Riemannian motion policies. arXiv preprint arXiv: 02854, 2018.
Ehresmann, C. Les connexions infinitésimales dans un espace fibré différentiable. In Colloque de Topologie, Bruxelles, volume 29, pp. 55-75, 1950.
Saviolo, A., Frey, J., Rathod, A., Diehl, M., and Loianno, Active learning of discrete-time dynamies for uncertainty- aware model predictive control. IEEE Transactions Robotics, 2023.
Gurriet, T., Mote, M., Singletary, A., Nilsson, P., Feron, E., and Ames, A. D. A scalable safety critical control frame- work for nonlinear systems. IEEE Access, 8:187249- 187275, 2020.
Hansen-Estruch, P., Shang. W., Pinto, L., Abbeel, P., and Tiomkin, S. Gem: Group enhanced model for learning dynamical control systems. arXiv preprint arXiv:2104.02844 0288444 2021.
Taylor, A., Singletary, A., Yue, Y., and Ames, A. Learning for safety-critical control with control barrier functions. In Learning for Dynamics and Control, pp. 708-717. PMLR, 2020.
Hsu, E. P. Stochastic analysis manifolds. Number 38. American Mathematical Soc., 2002.
Taylor, A. J. and Ames, A. D. Adaptive safety with control barrier functions. In 2020 American Control Conference (ACC), pp. 1399-1405. IEEE, 2020.
Jankovic, M. Robust control barrier functions for con- strained stabilization of nonlinear systems. Automatica, 96:359-367, 2018.
Vitelli, M., Chang, Y., Ye, Y., Ferreira, A., Wolczyk, M., Osiński, B., Niendorf, M., Grimmett, H., Huang, Q., Jain, A., et al. Safetynet: Safe planning for real-world self- driving vehicles using machine- learned policies. In 2022 International Conference on Robotics and Automation (ICRA), pp. 897-904. IEEE, 2022.
Kalman, R. E. A new approach to linear filtering and pre- diction problems. Journal of Basic Engineering, 82(1): 35-45, 1960.
Kobayashi, S. and Nomizu, K. Foundations of differential geometry, volume 2, volume 61. John Wiley & Sons, 1996.
Wabersich, K. P., Taylor, A. J., Choi, J. J., Sreenath, K., Tomlin, C. J., Ames, A. D.. and Zeilinger, M. N. Data- driven safety filters: Hamilton-jacohh reachability, control barrier functions, and predictive methods for uncertain systems. IEEE Control Systems Magazine, 43(5):137- 177, 2023.
Langley, P. Crafting papers on machine learning. Langley, P. (ed.), Proceedings of the 17th International Conference Machine Learning (ICML (CCM),000), pp. 1207-1216, Stan- ford, CA, 2000. Morgan Kaufmann.
Wu, X., Liu, K., and Shi, W. Structure algo- rithms for oscillatory differential equations II. Springer, 2015.
Lewis, A. D. Affine connections and distributions with applications to nonholonomic mechanics. Reports on
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Xian, Z., Gervet, T., Xu, Z.,, Qiao, Y.-.., Wang, T.-H., and Wang, Y. Towards generalist robots: A promis- ing paradigm via generative simulation. arXiv preprint 1rXiv23305.00455, 2023.
Zhang, B., Mao, Z., Liu, W. and Liu, J. Geometric rein- forcement learning for path planning of uavs. Journal of Intelligent & Robotic Systems, 77:391-409, 2015.

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Learn, Dyn, Environ. Constr. Meas--Indueed Bundle Struct.
A. Proof of Theorem 3.1
Proof. We provide a complete proof through six main steps:
Step 1: Preliminaries and Assumptions
First, recall the measurement-adapper Control Barrier Function (mCBF) conditions for the safe set So = fxr E EM ho(z))
0}. function & + R is an mCBF if
(i) b(x (())) >
I E So (safety implication)
(ii) 3a E Ee such that wnf + ab)) 2 0 (invariance condition)
(iii) V vV, w) L, for some Ls (Lipschitz measurement sensitivity)
Consider the system dynamics
i = f(x, u) + (())
(21)
y - h(x) +
with admissible noise sequences satisfying:
[|((t)] < 8 VI 0 (bounded process noise)
(22)
([]]=([)]| > 17) S exp( (xpp(11//(20))) ((2))) (sub-Gaussian measurement noise)
Step 2: Forward Invariance Under Perfect Mcasurements
Lemma A.1 (Nominal Safety). Given ((((0))(((())) 2 b0 0, under perfect measurements (e = 0), there exists bmnn > 0 such that
b(((1,, (((())) bmii::
11 2
(23)
Proof of Lemma A.I. By the mCBF condition (ii), one has
b ab())
(24)
The Comparison Lemma implies
b(((1), ((x()) 2 B(bo, 1)
(25)
where 3 is class KCC function. Take brvi infe20 B(b, t) 0.
Step 3: Uncertainty Propagation Analysis
Define the uncertainty tube:
T= ((x E E: : fyy --(x))l <
(26)
Lemma A.2 (State Uncertainty). Assume that z00) =(0) 0. For any trajectory with [[w(t)] s sw.
||x(() - a(1]l S Voo (6w)
(27)
where i(() is the nominal trajectory and The E Koo.
Proof of Lemma A.2. By Lipschitz continuity of and g. we have
| -l L 2lll + Lydw
(28)
Gronwall's inequality yields
Ln (e 1) A Tau (5m)
[ax(1) =(1)ll s
(29)
T,
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Learn, Dyn, Environ. Constr, Meas--Induced Bundle Struct.
Step 4: Local Safety Certification
Lemma A.3 (Safety Margin). At any time the safety margin satisfies:
b(x((),((t) byiin Lollu(t)]l 76(6w)
(30)
where To E Kix depends system parameters.
Proof of Lemma A.3. Decompose the safety margin:
b(x(1), = ((x(1),))))))))))())) h(x(())) + [6(x((,, v())
(31)
2 bmin Lollu(t)]l L6 (//1((a (6w)
under the condition (iii), where "o (6w) == (6w)
Step 5: Temporal Correlation Analysis
Define the safety violation event at time t:
A. = (b(((t(), v()) < 0
(32)
Lemma A.4 (Temporal Correlation). For any times 1,
P(A, n Av) S exp ( min c2/
(33)
where C1, > 0 are system-dependent constants.
Proof of Lemma A.4. By the Gaussian tail bound and temporal decorrelation of noise:
P(A, n Av) & P(]]((t)] > ne. ||l((t((]] > mh)
(34)
I exp minfcrle - L11, c2/ 6%.
where = (dmin (1
Step 6: Global Safety Guarantees
For any partition [0, oc) U (K 1)
P(3: 2 0: xt) & So)
= PPL 0: b(a(t). v(1) < 0)
P( P(ARs))
II -()-))) p.) 1=0
(35)
k=
exp(-c2/88)
expl- exp(-c
< ccp()
where Pj represents the probability that x(t) stays within the safety set in the j-th interval. C2 and At is chosen such that ei At 2 02 Therefore, we have
PP (P((1)) E So, 14 v22 2 1 exp(-c/88)
(36)
This completes the proof.
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B. Proof of Theorem 4.1
Proof. We prove this theorem in two parts:
First, we prove the convergence bound of learning dynamies: I flle S a exp(-A1t) + o2óy.
Consider the Lyapunov function candidate:
1 2 - -fll V(t)
(37)
2
where || ll denotes the norm induced by the metric on the fiber bundle E. Taking the derivative along the learning trajectories:
v(t)= = -.ñ
-f - f.L ff f
(38)
S -mmin (c,) - ffli llf - f
where ) is the minimum eigenvalue of the operator L1 (assume that all of its eigenvalues are real).
Applying Young's inequality yields:
Amin (C1)
8 1f f1 + 2mmin (11)
llf flle 6, 1
(39)
2
Therefore,
Amin (C1) ((t) + 2Amin (C1)
83
V(t)= <
(40)
By the comparison principle, we obtain
82
V(t) < V(0) exp() (0m0--mim(
(41)
X2 (L1)
Taking the square root and setting appropriate constants yields the first conclusion.
2) Next, prove the probabilistic safety guarantee: P(x((() S.) exp(-ca/88
Consider the measuremenn-adaaped safety certificate b: E - R, which by definition satisfies:
b(x, v)
2 E
(42)
For any trajectory (ar(), (t)), define the event: A, (see (32) for its definition).
Using the Lipschitz condition of mCBF and measurement noise bounds:
P(A,) ((z)(())(((())))))))) Mc((,
(43)
S cxpp-BB(((())
where Lg is the Lipschitz constant of the safety certificate with respect to measurements. By the properties of mCBF, there exists constant bmin such that:
Mx(t), 2 bynin
(44)
Setting C3 = (2L3) yields the second inequality.
Thus, we complete the proof the theorem.
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C. Additional Experimental Details
C.1. Performance Metries Details
We establish an evaluation framework with the following metrics:
The primary task completion metric is the success rate SR Nusos 100%, which measures the percentage of trials reaching
Nyaa
the goal state without constraint violations. Motion efficiency is characterized through the path length PL. = fo [()],,
the number of steps required GRS min(t [xx(t) Xgoal| S <}. and the final state error FSE |x -goaall
For safety evaluation, examine both instantaneous and aggregate constraint satisfaction across discrete time horizon 1 E (I,...,7) T). The minimum margin to constraints MMC = min, h (x()) captures the worst-case safety margin, while the average margin AMC = Σ h(x(t)) reflects the overall safety buffer maintained throughout the motion. The constraint satisfaction rate CSR = & 1t=1 I(h(x(t)) 2 0), where I(() is the indicator function, provides a statistical measure of safety performance.
The control quality is assessed through both magnitude and smoothness metrics. T Ei= [|u[t)]] measures resource efficiency, while control smoothness CS 7 E/= ||u(t + 1) - u(t)]l quantifies the continuity of the generated motion.
The average control magnitude ACM
C.2. Evaluation Protocol Details
For the worm robot (0.1m length), the initial position randomly sampled within a 0.4m x 0.4m region near the workspace origin, while the target position is sampled from similar-sized region distance of 0.6-0.8m. red spherical obstacle (radius 0.07m) is randomly placed between the start and goal positions. We generate 500 random trajectories, with success requiring reaching within 5em of the target while maintaining minimum 3cm obstacle clearance.
The Franka arm experiments consist of 400 point-to-point movements. Initial and target end-effector positions are randomly sampled from the reachable workspace. A red cubic obstacle (0. (0.1m x 0.1m x 0.1m) is randomly positioned at height (x=0.3++00.1 Im) between start and goal positions. Task completion requires the end-effector reaching within 2cm of the target position while maintaining 5cm minimum obstacle elearance.
For the quadrotor, test 300 navigation episodes. The start position is sampled near ground level (z=0.1 100005m), while goal positions are generated at varying heights (2=0.5-0.7m) within 0.5m radius. A red spherical obstacle (radius 0.07m) is randomly placed along potential flight paths. Success criteria include reaching within 10cm/5 of goal pose while maintaining 15cm safety margins.
C.3. Hardware and Software Configuration
Our experiments are conducted on a workstation equipped with an Intel Xeon CPU, NVIDIA RTX 3090 GPU (24GB GDDR6X), and 64GB DDR4 RAM. The software stack consists of Python 3.9 and Py 1.12.0, supported by CUDA 11.7 and cuDNN 8.5 for GPU acceleration. To ensure reproducibility, we fix random seeds to 42 across all experiments, controlling for randomness in PyTorch, NumPy, and environment initializations.
C.4. Reinforcement Learning Framework
Our SAC implementation follows the standard architecture with carefully tuned hyperparameters. The framework uses a discount factor 7 0.99 and a soft update coefficient of 0.005. The target entropy is set to the negative dimension of the action space, following common practice. All policy components (actor, critic, and entropy networks) use learning rate of 3 x 14 The replay buffer maintains 1 106 transitions, allowing for sufficient exploration while preventing overfitting to recent experiences.
C.5. Baseline Implementations
For model-based baselines, implement GPMPC using GPy'Torch with RBF kernels (length scale 1.0), which provides efficient Gaussian Process computations on GPU. RobustSafe employs tube MPC with prediction horizon of 20 steps and 0.05s sampling time, balancing computational efficiency with prediction accuracy. ALMPC utilizes the CasADi framework with IPOPT solver, configured for a maximum of 100 iterations per optimization step. Learning-based baselines maintain
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their original architectures while adopting our standardized training process for fair comparisons.
C.6. Neural Network Details
The neural network architecture consists of three hidden layers with 128, 64, and 32 units respectively, using ReLU activations throughout. Layer normalization is applied after each hidden layer to stabilize training. For barrier functions, we add tanh activation in the output layer ensure the boundedness of safety certificates. We enable mixed precision training to leverage the RTX 3090's Tensor Cores, and implement CUDA graph optimization for static computational graphs to maximize throughput.
C.7. Training Process
The training process employs the Adam optimizer with B1 0.9 and 32 0.999, coupled with cosine annealing learning rate schedule starting at 5 x 10-4. We use batch size of 256 to fully utilize the GPU memory while maintaining stable gradients. Gradient clipping with maximum norm of 1.0 prevents extreme parameter updates. Early stopping with a patience of 20 epochs prevents overfitting, and we maintain 20% validation split for monitoring training progress.
The training time varies across methods, with our approach requiring approximately hours, Neural-CBF 4 hours, and GPMPC hours the RTX 3090. Other baselines range from 3 hours depending on their computational complexity. To ensure statistical significance, we repeat all experiments 10 times with different random seeds, reporting mean performance metrics with standard deviations.
D. Additional Experiments on Broader Domains
D.I. Motivation and Dataset Analysis
To validate the broader applicability of our measurement-induued bundle structure framework, we first investigate three representative domains: building automation through the ASHRAE Building Operations Dataset, chemical process control using historical Industrial Batch Records (IBR), and power system management data from Regional Transmission Orga- nizations (RTO). These sources provide valuable insights into real-world measurement uncertainty patterns and system behaviors. Our analysis focuses on measurement characteristics that significantly impact safety-critical control decisions.
The building automation data reveals patterns in HVAC sensor networks, particularly regarding temperature and humidity measurement uncertainties across multiple zones. The IBR data demonstrates characteristic measurement delays and noise patterns in reaction vessel monitoring, especially for temperature and concentration measurements. The RTO data shows how grid frequency measurements are affected by network topology and communication infrastructure.
D.2. Environment Design Philosophy
Based on these domain insights, we design three simulation environments that preserve essential measurement uncertainty characteristics while enabling active control evaluation. Our design philosophy emphasizes fundamental physical principles and measurement challenges common in these domains, rather than replicating specific industrial configurations. This approach allows systematic evaluation of our framework's capability to handle different types of measurement uncertainties while maintaining safety guarantees.
D.3. Simulation Environment Details
The Building Climate Control environment models three-zone building with simplified thermal dynamics. The state vector T2, T2,T3, Ts. H1. H2, Hs) includes temperatures and humidity levels of each zone. The temperature dynamics follow -== EjeN, (Ty T.) + aTTams T,) + 3u,, where NY represents adjacent zones, kiz are heat transfer coefficients, and u; is the control input. Humidity follows similar mass transfer principles. Measurements include additive Gaussian noise V ~ (O, 2) and drift d(t) = At. Safety constraints maintain temperature between 20%C and C and humidity between 30% and 70% during occupied periods.
7: 7
The Batch Reaction Control environment implements single-vessel exothermic reaction. The state [T, CA, Cp) represents temperature and concentrations, following dynamics Î ki_ (T T.) + AH rT, CA) for temperature mep
op and ČA- Esp (CAo CA) (TTCCA) for reactant concentration, where r(T, CA) -oe--eee -../// C' is the reaction
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Learn. Dyn, Environ. Constr. Meas--Inuuced Bundle Struct.
rate. Control inputs are composed of cooling temperature T. and feed rate Fin. Measurements include Gaussian noise with standard deviation of O.3%C for temperature and 0.02 mol/I. for concentration, plus 5-second delay modeled as (h((z) 2) + v. Safety constraints maintain temperature below C and ensure minimum product quality.
The Grid Frequency Control environment represents four-node power network where each node's state includes frequency deviation Af; and power imbalance A P. The frequency dynamics follow the simplified swing equation Af; = 2Hr, (AP; D. - Ejev; Ba, (AO, - A0,)), where H; is inertia constant and D; is damping. Control actions adjust power generation rates within +50 kW/s. Frequency measurements include noise v ~ (0, 0.01²) Hz and 100ms delay, expressed as y (t) ff - Ta) + (t).). Safety requirements maintain frequency deviations within +0.5 Hz.
These environments employ basic numerical integration with 0.e-sec time steps. The simplified dynamics capture fundamental behaviors while maintaining essential characteristics of measurement-basedd safety control: state estimation under uncertainty, coupled dynamics between subsystems, and constraint satisfaction with imperfect information. This basic implementation provides clear validation of our framework's 's core capabilities while ensuring the reproducibility of results.
D.4. Experimental Setup
Each environment runs for 1000 episodes with randomized initial conditions and disturbance patterns. We implement the environments using Python with standard numerical libraries. The sampling rates set according to domain characteristics: 1-minute intervals for building control, 30-second intervals for batch reactions, and 20ms intervals for grid frequency control.
Performance evaluation uses consistent metrics across all domains: Success Rate (SR) measures task completion while maintaining safety constraints, Constraint Satisfaction Rate (CSR) tracks the percentage of time safety constraints are satisfied, Average Control Magnitude (ACM) measures control effort, and Control Smoothness (CS) evaluates control stability. All experiments are repeated 10 times with different random seeds to ensure statistical significance.
D.5. Results and Analysis
Table 3 presents the comparative performance results across different domains and methods.
Table Cross-Domain Performance Comparison
Domain
Method
SR (%)
CSR (%)
ACM
CS
Building
Ours
95.3+111
99.4+0.2
0.14#0.02
0.02+0.003
Control
Neural-CBF
83.6+1.8
97.6+0.4
0.22+0.03
0.04+0.005
GPMPC
79.2+2.0
98.8+0.3
0.25+0.03
0,05#0.006
Batch
Ours
94.1#1.2
99.1
0.1520.02
0,03+00.004
Reaction
Neural-CBF
82.3+1.9
97.2+0.5
0.23+0.03
0.05+00006
GPMPC
78,5+2.1
98.5+0.4
0.27#0.04
0.06+00007
Grid
Ours
93.8+1.3
98.9+0.3
0.1620.02
0.03+0.004
Frequency
Neural-CBF
81.9+1.8
96.8+0.5
0.24+0.03
0.05+0.006
GPMPC
77.642.2
98.3+0.4
0.28+0.04
0.06+00007
The results demonstrate consistent superior performance of our method across all domains. In building control, our approach achieves a 95.3% success rate while maintaining tight comfort constraints, significantly outperforming baseline methods. The high constraint satisfaction rate (99.4%) indicates robust handling of sensor drift and inter-zone coupling uncertainties.
For batch reaction control, our method shows strong performance with 94.1% 1 success rate despite challenging measurement delays and process uncertainties. The framework effectively balances product quality constraints with operational safety bounds, requiring lower control effort (ACM = 0.15) compared to baseline approaches.
In grid frequency control, the method maintains reliable performance (93.8% success rate) while handling both commu- nication delays and measurement uncertainties. The framework's ability to adapt to measurement quality variations is particularly evident the high constraint satisfaction rate (98.9%) achieved with smooth control actions (CS = 0.03).
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D.6. Discussion
These results validate our framework's effectiveness across fundamentally different physical domains with varying uncer- tainty characteristics. The consistent performance advantages over baseline methods suggest that the measurement-induueed bundle structure provides general approach for handling diverse measurement uncertainties in safety-critical control applications.
Particularly noteworthy is the framework's ' ability to maintain high performance across different temporal scales and physical processes. This adaptability, combined with the framework's 's capability to handle both systematic and random measurement uncertainties, demonstrates its potential for broad application in real-world control systems.
The lower control magnitudes and higher smoothness metrics achieved by our method indicate more efficient control strategies that better account for measurement uncertainty characteristics. This efficiency likely stems from the framework's s geometric treatment of measurement uncertainty, which enables more informed decisions about when and how to apply control actions based on measurement quality.
The results also highlight the framework's ability to balance multiple competing objectives: maintaining safety constraints, achieving control objectives, and minimizing control effort. This balance achieved consistently across domains with different physical characteristics and measurement challenges, suggesting the fundamental soundness of our geometric approach uncertainty handling.
E. Further Ablation Studies
To provide deeper insights into our framework's behavior, we conduct comprehensive ablation experiments from three perspectives: measurement uncertainty characteristics, bundle structure variations, and comparison with alternative geometric representations. These experiments systematically evaluate how different components contribute to the framework's overall performance and robustness.
E.1. Impact of Measurement Uncertainty Patterns
We first investigate how different measurement uncertainty patterns affect system performance. Beyond the standard Gaussian noise assumption, we examine various uncertainty types commonly encountered in practical applications. Table presents the comparative results across different uncertainty patterns while maintaining consistent control parameters.
Table 4. Performance Under Different Mcasurement Uncertainty Patterns
Uncertainty Type
SR (%)
CSR (%)
ACM
CS
Gaussian (o=0.1)
9.1
99.1
0.17-00.2
0.03110.004
Gaussian (a=0.3)
9..8+11.3
98.5
0.19-10.02
0.0450.005
Fixed Bias (0.2)
9.2
98.7+0.3
0.18
0.03
Time- Varying Bias
91.9+114
98.2+0.3
0.20+10.03
0.04
50ms Delay
94.1
98.910.22
0.18:10.02
0.03110000
100ms Delay
92.3#1.3
98.4+0.3
0.19=0.02
0.04
Sensor Failure
89.7#1.5
97.8+10.4
0.22-10.03
0.05400006
The results reveal that our framework maintains robust performance across various uncertainty patterns, with success rates consistently above 89%. While performance slightly degrades with increasing noise magnitude, the degradation is gradual and predictable. Notably, the framework shows particular resilience to measurement delays up to 50ms, maintaining a 94.1% success rate with minimal impact on control smoothness.
E.2. Bundle Structure Variations
We next examine how different design choices in the bundle structure affect system performance. We implement and evaluate four variations of the bundle structure: fixed dimension, adaptive dimension, simple connection, and complex connection. Table 5 summarizes the comparative performance metrics.
The adaptive dimension variant demonstrates superior performance across all metrics, achieving a 94.8% success rate and
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Table 5. Performance Comparison of Bundle Structure Variations
Bundle Variation
SR (%)
PL (m)
GRS
MMC (m)
CSR (%)
ACM
Fixed Dimension
20.3:108 9..1=1.3
412=119
0.23
0.19:00.0 98.511..3
Adaptive Dimension
94.8+1.1
19.1
395=177
0.24:1003
99.1=0.2
0.18+0.02
Simple Connection
90.5+11.4
21.2++0.
428:220
0.22
98.1+0.33
0.20+10.03
Complex Connection
93.7+11.2
19.8+0.8
405+18
0.24+
98.8+0022
0.18+
reduced path length of 19. Im. This improvement can be attributed to its ability to adjust the bundle structure based on local measurement quality and system state. The complex connection variant also shows strong performance, particularly in maintaining higher safety margins (MMC = 0.24m) compared to simpler alternatives.
E.3. Necessity of Fiber Bundle Structure
To demonstrate why fiber bundle structure is essential for unifying measurements and safety constraints, we conduct systematic ablation experiments focusing on the geometric framework's 's key components and their interactions with measurement uncertainty.
Table 6. Comparison of Geometric Representations for Measurement Uncertainty
Geometric Structure
SR (%)
PL (m)
GRS
MMC (m)
CSR (%)
ACM
Fiber Bundle (Ours)
96.3+1.1
18.5+0.7
3831:18
0.24
99.3:1022
0.17:0..2
Product Manifold
1.4
2..3:10.9
472::21
0.16
94.2.10.3
0.28:10.03
Vector Bundle
89.8+1.3
23.2+008
445+=20
0.18++0.03
95.8++0.3
0.25++0.03
Principal Bundle
90.4
22.41:08
4281:9
0.19::0.03
96.1
0.23
The fiber bundle structure fundamentally differs from alternative representations in its intrinsic separation between base space dynamics and measurement uncertainties in fiber directions. This geometric decoupling enables precise uncertainty propagation while maintaining the underlying dynamical structure, leading to significantly improved success rates (96.3% vs 87.5-90.4%) and constraint satisfaction (99.3% vs 94.2-96. 1)). Unlike product manifolds that treat state and measurement spaces independent entities, or vector bundles that lack natural parallel transport, fiber bundles provide canonical vertical-horizontal decomposition that directly captures measurement-statee relationships.
The superior performance results from two key theoretical properties: First, the fiber bundle's connection form automatically adapts safety constraints based on local measurement quality, enabling tighter safety bounds (MMC 0.24m vs 0.16-0.19 without conservative over-approximation. Second, the bundle projection naturally maintains consistency between local measurement-sppac behaviors and global state-space trajectories, resulting in significantly shorter paths (18.5m vs 22.4- 25.3m) and more efficient control actions (ACM 0.17 vs 0.23-0.28). These geometric advantages make fiber bundles not just mathematical convenience but a necessary structure for properly unifying uncertain measurements with safety-critical control.
F. Discussion on Comparisons with Recent Methods
While direct numerical comparisons with recent geometric deep learning approaches may be natural, there are several fundamental differences that make such direct benchmarking potentially misleading. Here we elaborate on why our fiber bundle framework occupies distinct theoretical niche and requires different evaluation criteria.
First, recent geometric deep learning methods like equivariant networks and manifold learning primarily focus repre- sentation learning and feature extraction while preserving geometric structures. For example, SEE(3) networks ensure rotational and translational invariance in learned features, and Riemannian VAEs learn manifold embeddings of data. In contrast, our framework explicitly models the geometric relationship between states and measurements, with safety guarantees as a primary objective rather than just geometric feature learning.
The crittal nature of our framework introduces fundamentally different requirements. While geometric deep learning
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Learn. Dyn, Environ. Constr, Meas--Inuceed Bundle Struct.
methods optimize for statistical performance metrics (e-g., reconstruction error, classification accuracy), our method must provide deterministic safety guarantees under measurement uncertainty. This fundamental difference in objectives statistical performance versus provable safety makes direct performance comparisons less meaningful.
Consider concrete example: an equivariant network might learn excellent state representations for robot manipulation, but it cannot inherently guarantee safety constraints under sensor noise. Our fiber bundle framework, through its explicit modeling of measurement uncertainty via fibers æ -t (a) and safety map & : E + R, provides these guarantees by construction. The trade-off between control performance and safety margins is explicitly encoded in our geometric structure.
Moreover, our framework's 's treatment of measurement uncertainty is fundamentally different. While existing methods might handle uncertainty through probabilistic embeddings or uncertainty quantification in learned features, our approach captures the intrinsic geometric relationship between measurements and states through the bundle structure. This allows us to maintain safety guarantees even when measurements degrade- a crucial requirement for real-world robotic systems that cannot be addressed solely through improved representation learning.
The computational objectives also differ significantly. Geometric deep learning methods typically optimize end-to-end performance on specific tasks, while our framework must solve constrained optimization problems that maintain safety invariants:
min ||u - Unoma well
(45)
subject to f, v) + abb2,, v)) 2
where b(x,)) y) is the measurement-awarr barrier function. This fundamental difference in problem formulation means that standard benchmarking metrics may not capture the essential anfett-critical concerns of our approach.
A more meaningful evaluation framework should instead focus on several key aspects of safety-critical control under uncertainty. The system's 's ability to maintain robust safety guarantees as measurement uncertainty varies, which demonstrates the fundamental reliability of our geometric approach. This includes testing performance under different noise levels, sensor failures, and measurement degradation scenarios.
The framework's 's flexibility in incorporating diverse safety constraints while preserving its geometric structure is another crucial evaluation criterion. This involves demonstrating how different types of safety requirements from simple collision avoidance to complex multi-objective constraints can be naturally encoded within the fiber bundle structure while maintaining theoretical guarantees.
Computational efficiency in safety-constrainee control synthesis represents another key evaluation dimension. The framework must demonstrate real-time performance in generating safe control inputs, particularly important for high-dimensional robotic systems operating in dynamic environments. This includes analyzing computational scaling with system complexity and the efficiency of our geometric optimization approaches.
Finally, the framework's 's generalization capabilities across different robotic platforms and sensor configurations provide a crucial measure of its practical utility. This involves demonstrating how the same geometric principles can be applied to diverse robotic systems- from manipulators to mobile robots while maintaining safety guarantees under different sensing modalities and control architectures.
These evaluation criteria better reflect the unique contributions of our fiber bundle framework, focusing on its core strengths in providing geometric safety guarantees under uncertainty rather than attempting direct comparisons with methods designed for different objectives. This approach allows us to properly assess the framework's effectiveness in bridging the gap between theoretical safety guarantees and practical robotic implementation.
G. Geometric Interpretation and Physical Insights
The fiber bundle framework provides a natural geometric structure for unifying measurement uncertainty with safety-critical control. The fundamental geometric objects consist of a base manifold M representing the true state space (e.g., robot configurations), a total space combining states with measurements, and fibers x- (xx above each state containing all possible corresponding measurements. The bundle projection :: E MM maps measurements to their underlying states, while the connection form w describes how uncertainty propagates through the system.
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For concrete intuition, consider drone equipped with depth sensors. The fiber above each position p R contains the set of possible depth measurements {y = h(p) + ||a||| S 6), where h represents the ideal measurement model and captures bounded uncertainty. The horizontal lift of trajectories maintains consistency between dynamics-predicted positions in the base space and actual sensor measurements in the fiber space, while respecting safety constraints through 4: E + R.
This geometric structure directly captures the physical reality of robotic systems. The state-dependent measurement quality naturally represented through varying fiber dimensions and geometry across the state space. For instance, in robotic manipulation, joint angle sensing accuracy often depends on the arm configuration q E Q. captured by the fiber structure variation over the configuration manifold. The connection form w ensures consistent propagation of these measurement uncertainties while respecting the underlying physical dynamics.
The bundle structure preserves key physical properties: mechanical energy conservation through the symplectic structure, and sensor-state correlations through the horizontal distribution. This enables measurementt-oonsistee dynamics learning via constrained neural ODEs:
-- fiter (V
(46)
The measurement-awwrc control barrier functions (mCBFs) automatically adapt safety margins based on measurement confidence:
b(x, y) = b0( h(x)) Lolly h(x)
(47)
key strength of this geometric approach lies in its coordinate independence and preservation of physical symmetries. The intrinsic nature of the bundle structure means the framework applics equally well across different robot representations while automatically preserving important physical invariants like SF) symmetry. This enables broad generalization across diverse robotic systems and tasks without requiring task-specifi modifications to the underlying mathematical framework.
The fiber bundle structure provides robust safety guarantees through its geometric mechanisms. The bundle projection ensures safety constraints are satisfied for all possible measurements within each fiber, while the connection form enables consistent propagation of these constraints as the system evolves. This geometric approach to safety fundamentally different from traditional methods using uncertainty ellipsoids tubes, it captures the intrinsic coupling between measurements and dynamics that naturally arise in physical systems.
While the framework requires sufficient smoothness of the underlying manifold M and bounded measurement uncertainty, these requirements align well with numerous practical dynamics (for example, robotic systems). The geometric decom- position into horizontal and vertical components often leads to efficient computations despite the additional dimensions introduced by the bundle structure. Future extensions could enable learning optimal fiber structures from data or handling coupled uncertainties in multi-agent systems while maintaining the core geometrie principles that provide rigorous safety guarantees.
This geometric perspective offers both theoretical guarantees and practical insights for implementing robust safety-critical control under uncertainty. By providing unified treatment of measurement uncertainty that respects physical constraints and symmetries, the framework bridges the gap between theoretical safety guarantees and practical implementation. The natural handling of both physical constraints and measurement uncertainty through intrinsic geometric structures makes it particularly valuable for real-world applications where ensuring safety under imperfect sensing is crucial.
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Reviews:
[{"review_id": "hkRaw77uND", "reviewer": "Reviewer_bzHQ", "summary": "This study proposes a geometric approach to learning dynamics with safety guarantees, leveraging the bundle structure to account for uncertainties. After presenting the geometric approach for controlled dynamical systems, the study introduces measurement-adapted control barrier functions, which enable the development of safety guarantees. Furthermore, the paper presents a learning framework for policy design with safety guarantees based on the bundle framework.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank you for your thoughtful feedback and have prepared appropriate revisions.\n\n## On Theorem 3.1 Formulation\n\nWe appreciate that you correctly noted that the theorem should explicitly specify the control law. We will revise Theorem 3.1 to clearly indicate that there exists a control strategy $\\pi(x,y) \\in \\Pi_{\\text{safe}}$ for which the safety probability bound holds. Specifically, $\\Pi_{\\text{safe}} =$ \\{$\\pi : M \\times Y \\to U \\mid L_e b(x,y,\\pi(x,y)) + \\alpha(b(x,y)) \\geq 0, \\forall(x,y) \\in E$\\}, where $L_e b$ is the extended Lie derivative along the fiber bundle connection, defined as $L_e b(x,y,u) = \\nabla_x b(x,y)^T f(x,u) + \\nabla_y b(x,y)^T \\frac{\\partial h}{\\partial x}f(x,u)$ in Section 3.6. This clarifies that safety guarantees apply to any control strategy respecting the mCBF condition.\n\n## Clarification of Contributions in Section 3\n\nThe system model in Section 3.1 builds on standard stochastic control formulations, but our innovation lies in integrating it with the fiber bundle perspective specifically for measurement uncertainty. Section 3.2's fiber bundle framework represents our original contribution, particularly the connection form in Equation (3) that couples measurement uncertainty with state dynamics through $K(x)$. To our knowledge, applying fiber bundles to measurement uncertainty in safety-critical learning is novel.\n\n## Time Setting Consistency Issue\n\nRegarding the apparent inconsistency between continuous-time dynamics and discrete-time cost, we can formalize this transition as follows: Define continuous-time cost $J_c(\\Theta) = \\int_0^\\infty e^{-\\rho t} c(x(t),\\Theta(x(t)))dt$ and discrete-time cost $J_d(\\Theta) = \\sum_{k=0}^\\infty \\gamma^k c(x(k\\Delta t),\\Theta(x(k\\Delta t)))$, where $\\gamma = e^{-\\rho \\Delta t}$. For small $\\Delta t$, we have $|J_c(\\Theta) - J_d(\\Theta)| \\leq C \\cdot \\Delta t$. Similarly, if continuous-time systems satisfy the mCBF condition $L_e b(x,y,\\pi(x,y)) + \\alpha(b(x,y)) \\geq 0$, discrete systems maintain safety under $(b(x_{k+1},y_{k+1}) - b(x_k,y_k))/\\Delta t + \\alpha(b(x_k,y_k)) \\geq \\beta \\cdot \\Delta t$, where $\\beta$ depends on second-order derivatives of system dynamics.\n\n## Clarification of Problem Setting\n\nTraditional methods use the separation principle: estimate state $\\hat{x} = E[x|y_{1:t}]$ then apply controller $u = \\pi(\\hat{x})$. Under high uncertainty, estimation error $e = x - \\hat{x}$ may violate safety constraints. Our approach constructs a safety framework directly on fiber bundle $E = M \\times Y$ with control strategy $\\pi : M \\times Y \\to U$, incorporating measurement uncertainty into the safety condition: $\\nabla_x b(x,y) \\cdot f(x,\\pi(x,y)) + \\nabla_y b(x,y) \\cdot \\dot{h}(x) + \\alpha(b(x,y)) \\geq 0$. This avoids intermediate state estimation and automatically adjusts safety margins with measurement quality—as $\\|y - h(x)\\|$ increases, the control becomes more conservative.\n\n## Concrete Examples of Learning Operators\n\nFor dynamics learning operator $L_1$ in (12), we implement it on a 2D system where $x = [p, \\xi]^T$, $f(x,u) = [\\xi, u]^T$, and observation $y = h(x) + w$ with $y \\in \\mathbb{R}^m$. Here, $h: \\mathbb{R}^2 \\rightarrow \\mathbb{R}^m$ is the observation function. The bundle structure comes from the relationship between states and measurements:\n\n$\nL_1(f_{\\theta} - f)(x,y) = \\nabla_x(f_{\\theta} - f) \\cdot f(x,u) + K(x)(y - h(x))\n$\n\nwhere $K(x) \\in \\mathbb{R}^{2 \\times m}$ is the gain matrix. This operator works along two directions: horizontally through state dynamics (first term) and vertically through measurement fibers (second term). In implementation, the parameter update is:\n\n$\\frac{d\\theta}{dt} = -\\Gamma \\sum_{i} [(f_{\\theta}(x_i) - \\dot{x}i)^T \\Sigma_i^{-1} \\frac{\\partial f_{\\theta}}{\\partial \\theta} + \\lambda K(x_i)(y_i - h(x_i))^T \\frac{\\partial f_{\\theta}}{\\partial \\theta}]$\n\nSimilarly, safety certificate operator $L_2$ in (12) for constraint $h_s(x) = x_{safe} - p \\geq 0$ becomes:\n\n$L_2(\\Phi - \\Phi^*)(x,y) = \\nabla_x (\\Phi - \\Phi^*) \\cdot f(x,u) + \\alpha ‖y - h(x)‖^2 \\nabla_y(\\Phi - \\Phi^*)$\n\nwhere $h_s: \\mathbb{R}^2 \\rightarrow \\mathbb{R}$ is a scalar safety constraint on position. This approach automatically adapts safety margins with measurement uncertainty. Consider position control with noisy sensors: traditional methods first estimate position and then control (introducing compounding errors), while our operators $L_1, L_2$ directly map measurements to controls, becoming more conservative as uncertainty increases through terms like $\\alpha‖y-h(x)‖^2$. Our 2D navigation system benchmark in the revision confirms this advantage: with noise $\\sigma_y=0.1$, dynamics learning error improves by 47% (0.08 vs 0.15); at $\\sigma_y=0.5$, improvement reaches 60% (0.17 vs 0.42). For safety near obstacle at $x_{obs}=2$ with $\\sigma_y=0.5$, our method maintains a 0.47 margin while standard CBFs drop to 0.22 with violations - showing how Equation 12 handles measurement uncertainty without separate estimation steps.", "claims_and_evidence": "The safety guarantee is given in Theorem 3.1. The numerical experiments show superior performance compared to existing methods.", "methods_and_evaluation_criteria": "The proposed method is based on a mathematical geometric framework, specifically leveraging bundle structures. The theoretical foundation is sound. I do not see any issues with the evaluation criteria for the numerical experiments.", "theoretical_claims": "As far as I have checked, the proofs in the appendix are sound. However, the statement of Theorem 3.1 may need to be revised. Since the control u is considered in the controlled dynamical system, the theorem’s statement should be like as follows:\n\nTheorem 3.1. Given an $m C B F$ b satisfying the preceding conditions, if $b(x(0), y(0)) \\geq 0$, then for any admissible noise sequences $w(\\cdot), v(\\cdot)$, **there exists $u_t$ ($t \\in [0, \\infty)$)**:\n$$\n\\mathbb{P}\\left(x(t) \\in \\mathcal{S}_0 \\text { for all } t \\geq 0\\right) \\geq 1-\\exp \\left(-c / \\delta_v^2\\right)\n$$\nwhere $c>0$ is a constant depending on system parameters.\n\nThe solution $x(t)$ depends on the control $u$. Therefore, the current form of the statement of Theorem 3.1 does not clearly specify which control $u$ is used to obtain the probabilistic bound.", "experimental_designs_or_analyses": "There are no issues with the experimental design or analyses.", "relation_to_broader_scientific_literature": "This study is motivated by the geometric approaches in analytical mechanics and control theory. The bundle plays an important role in these fields. In this sense, a key contribution of this study—introducing the bundle-based approach—is its role in bridging the existing results in control theory and learning.", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "This paper has its strength in proposing a geometric approach based on the bundle structure. This is interesting, but it may be necessary to improve the presentation to ensure that readers clearly understand the contribution. Specifically, it is somewhat unclear whether all the content in Section 3, such as the modeling in 3.1 and the fiber bundle framework in 3.2, is proposed in this study or has already been presented in existing papers.", "other_comments_or_suggestions": "In Section 4.4, the cost $J(\\Theta)$ is given in a discrete-time setting. However, the system in (1) is formulated as a continuous-time system. Please clarify this inconsistency.", "questions_for_authors": "1. The problem setting is somewhat unclear. Given the system in (1), which includes the measurement equation, a typical setting would be a partially observed control setting. However, the policy in Section 4.4 is given in the form of $\\pi_{\\Theta}: \\mathcal{M} \\to \\mathcal{U}$, which has access to the value of $x$. It is unclear why the measurement y is considered when obtaining the safety policy.\n2. I was wondering if it is possible to provide a concrete example of $\\mathcal{L}_1$ and $\\mathcal{L}_2$ in (12). I could not fully follow the discussion in Section 4.1.", "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": "LO7F9dStxI", "reviewer": "Reviewer_xLW4", "summary": "This paper considers the problem of learning unknown dynamics models in the presence of model constraints. The paper points out that classical treatments of this problem ignore the system's inherent geometry while taking measurements into account and, in doing so, ignore important information that could be useful during learning. However, measurement uncertainty induces a fiber bundle structure which naturally lends itself to neural ODE models. Positive results are shown in an extensive empirical study.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your evaluation and suggestions. Your positive assessment is encouraging, and we accept the improvements you suggested. Below are the enhancements we plan to implement:\n\n## Correction of Theoretical Statements\n\nWe will add the dependency on $\\alpha$ in equation (9), correcting it to $\\inf_{u\\in U}[L_fb + L_gbu + \\alpha(b)] \\geq 0$ to ensure mathematical rigor. In the statement of Theorem 4.1, we will specify the conditions for the real-valued eigenvalue assumption. We will add: \"Assume that $\\mathcal{L}_1$ is a symmetric positive definite operator\" (so it only has positive eigenvalues) and discuss in which practical systems this assumption naturally holds, such as in mechanical systems with physical dissipation processes.\n\n## Enhancing Accessibility of Technical Content\n\nWe will make the technical content accessible by enhancing the intuitive explanation of mCBF conditions. Condition 1 ($b(x,y) \\geq 0 \\Rightarrow x \\in S_0$) ensures that positive values of the safety certificate directly correspond to physically safe system states; Condition 2 ($\\inf_{u\\in U}[L_fb + L_gbu + \\alpha(b)] \\geq 0$) guarantees that system dynamics automatically \"avoid\" unsafe region boundaries, with stronger \"avoidance force\" as boundaries are approached; Condition 3 ($|b(x,y_1) - b(x,y_2)| \\leq L_bd_Y(y_1,y_2)$) ensures that measurement noise does not cause drastic changes in safety certificates, providing robustness against measurement uncertainty. We will add a key visualization showing how trajectories in the state space M intersect with measurement uncertainty fibers in the total space E, and how the connection form transfers tangent vectors from the base space to the fibers, demonstrating the difference between our method and traditional approaches in handling measurement uncertainty.\n\n## Key Insights on Fiber Bundle Structure for Measurement Uncertainty\n\nTraditional methods and our approach have differences: traditional methods treat measurement uncertainty as external disturbances, requiring first estimating states then controlling (introducing compound errors). In contrast, our method treats measurement uncertainty as an intrinsic geometric property of the system structure, directly mapping from measurement space to control actions through the fiber bundle structure. We will add visualization of the fiber bundle structure, showing how state trajectories in the base space M generate \"measurement uncertainty tubes\" in the total space E, and how the connection operator $K(x)(y-h(x))$ automatically adjusts safety boundaries. We will explain how the terms $K(x)(y-h(x))$ and $\\alpha\\|y-h(x)\\|^2\\nabla_y\\Phi$ in equation (12) act as natural \"regulators\" of measurement quality, automatically increasing conservatism in uncertain regions. These terms transform the geometric structure of measurement uncertainty into adaptive safety margins, enabling our method to enhance safety guarantees as measurement uncertainty increases.\n\n## Improving Fairness and Relevance of Experimental Analysis\n\nRegarding the comparison with physical consistency baseline methods, we will articulate the fairness of comparisons. The PNDS method focuses on maintaining physical consistency but lacks explicit handling of measurement uncertainty; GeoPath and GEM excel at maintaining geometric invariance, but they were not designed to handle uncertainty induced by measurements. We will add detailed quantitative comparative analysis, including performance of different methods under various noise levels, changes in physical constraint violation rates as measurement noise increases, and the relationship between dynamic learning errors and measurement uncertainty. We will explain how we ensure all methods use the same neural network architecture, computational budget, initial and target state distributions, noise distributions, and system parameter uncertainties, providing a fairer basis for comparison.\n\n## Integration of Recommended References\n\nWe will create a new subsection discussing connections with sensor geometry learning research. Pierce & Kuipers' work on constructing spatial embeddings from sensor correlations provides interesting parallels to our fiber bundle structure, though our focus extends to safety guarantees under uncertainty. Their work on learning hierarchical models from uninterpreted sensorimotor signals also resonates with our approach to learning dynamics under environmental constraints. We will explore how these perspectives complement our measurement-adaptive learning methods and discuss how our framework might benefit from their insights on sensor space embedding and abstraction of continuous environments to more manageable representations.\n\nWe thank you for the valuable suggestions, which help improve our work and provide guidance for our research direction. We believe that with these improvements, the paper will be clearer, more rigorous, and more accessible while maintaining its technical depth and theoretical contributions.", "claims_and_evidence": "*Claims*\n1. Proposes a novel geometric framework that unifies measurement uncertainty, system dynamics, and constraints within fiber bundle structures\n2. Introduces adaptive measurement aware safety certificates that adjust conservative margins based on local measurement quality.\n3. Demonstrates enhanced generalization capabilities across different scenarios without requiring global information through experimental validation.\n4. The proposed method is theoretically sound; learning converges with a certificate of safety. \n\n*Evidence*\n1. This is supported with the formalism in section 3. Further support comes from the context provided by related work.\n2. This is also supported with the formalism in section 3. \n3. Experiments with three simulated environments provide the data supporting this claim. The proposed method consistently performs well when compared with a variety of metrics and among relevant state-of-the-art baseline methods. \n4. Evidence comes from Theorems 3.1 and 4.1.", "methods_and_evaluation_criteria": "Experiments evaluate the proposed method in three simulation environments. \n1. Worm robot\n2. Manipulator arm\n3. Quadrotor drone\n\nUniformly positive results among these suggest an algorithm's ability to perform well across a variety of settings.\n\nPerformance is evaluated with\n\n- Success rate\n- Path efficiency measures\n- Safety margins  \n- Control quality\n\nWithin these settings, experiments consider four kinds of evaluations. \n\n*Learning-based Safety Certification:* Methods with fixed barrier functions achieve reasonable success but come second to the proposed method.  \n\n*Physics-informed and Geometric Methods:*  The proposed method outperforms these baselines in this environment, but the paper does not explain why. The baseline was introduced as a check of \"physical consistency.\" What that means precisely is left unsaid. Further explanation of this result is needed to avoid the baseline being viewed as a strawman. \n\n*Robust and Adaptive Control:* Baselines achieve high rates of constraint satisfaction while producing overly conservative trajectories. The proposed method avoids this pathology. \n\n*Uncertainty-Aware Predictive Control:* The proposed method ranks highest under this evaluation due to its integration of relevant measurement uncertainty with constraints.  \n\nFinally, the paper performs an ablation of several aspects of their method to demonstrate the relative performance gain of each feature.\n\n\n*Assessment :* Overall the experiments were thorough, comprehensive, well-documented, of sound methodology, and demonstrate consistently positive results.", "theoretical_claims": "*Claims*\n\nThe paper makes two main theoretical claims.\n\n1. Theorem 3.1 (paraphrased): Whenever the system starts under safe conditions, the probability that the state remains safe is bounded below. \n2. Theorem 4.1 (paraphrased): The proposed learning dynamics converge in a safe manner. \n\n*Evidence*\n1. The proof of this involved several intermediate results. Each was presented clearly and appears sound.  \n2. The proof of this was easy to follow and appears sound. The proof makes an assumption about real-valued eigenvalues. The conditions under which this assumption hold should qualify the theorem statement. \n\n*Minor comment about the formalism*\n- The dependence of $u$ is missing from (9).", "experimental_designs_or_analyses": "I thoroughly reviewed the experiments, their methodology, results, and analysis. More comments can be found in my response to methods.", "relation_to_broader_scientific_literature": "This paper is well-positioned with respect to related work. The paper does a good job of clearly covering the broader areas of research which it intersects, and providing descriptions of more closely-related works.", "essential_references_not_discussed": "I did not identify any essential references that were omitted. \n\n\nBelow are some pointers to non-essential related work---references the authors may find interesting.\n\n*Learning sensor geometry*\n1. [Map learning with uninterpreted sensors and effectors](https://www.sciencedirect.com/science/article/pii/S0004370296000513)\n2. [Discovering sensor space: Constructing spatial embeddings that explain sensor correlations](https://ieeexplore.ieee.org/document/5578854)\n3. [Adapting the Function Approximation Architecture in Online Reinforcement Learning](https://arxiv.org/pdf/2106.09776)", "other_strengths_and_weaknesses": "*Strengths*\n- This paper checks about every box \n- Originality: The paper appears meaningfully novel, both theoretically and from a practical perspective. \n- Clarity: The paper is exceptionally clear. I expect that even non-experts from this area will be able to grasp its main contributions.\n- Significance: Based on the theoretical results and the extensive empirical validation, I think it's fair to say this paper makes a significant contribution toward algorithms that learn how systems move and behave, while ensuring they stay safe, even when they only have limited information about their surroundings.\n\n*Weaknesses*\n- The material is quite technical and, as a result, unnecessarily restrict its potential audience. It may be possible to appeal to more readers by elaborating on key concepts, such as the three mCBF conditions.", "other_comments_or_suggestions": "I have no additional comments to add here.", "questions_for_authors": "I have no questions for the 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": "m6uljPqMPl", "reviewer": "Reviewer_Bp16", "summary": "This paper presents a novel geometric framework for learning unknown dynamics under environmental constraints when constraint information is only locally available and uncertain. The authors introduce a fiber bundle structure over the state space that unifies measurements, constraints, and dynamics learning. This geometric approach enables measurement-aware Control Barrier Functions (mCBFs) that adapt to local sensing conditions. By integrating Neural ODEs, the framework learns continuous-time dynamics while preserving geometric constraints, with theoretical guarantees of learning convergence and constraint satisfaction dependent on sensing quality. The authors demonstrate through simulations that their approach significantly improves both learning efficiency and constraint satisfaction compared to traditional methods, especially under limited and uncertain sensing conditions.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your feedback.\n\n## Experimental Details & Baselines\n\nOur Genesis physics engine uses task-specific configurations: Semi-implicit Euler (5e-4s timestep) with neo-Hookean material (μ=2kPa, λ=10kPa) for soft worm; RK4 (1e-2s timestep) with friction coefficients (0.15 static, 0.09 dynamic) for Franka; and RK4 (2e-3s timestep) with aerodynamic effects for quadrotor. Simulations employ continuous collision detection (3mm threshold) and sequential impulse solving (200 iterations). Statistical validation includes 95% CIs from 10 trials, with variation coefficients of 3.2-6.7%. We use Poisson disk obstacle sampling (20cm separation), bi-directional RRT for feasibility checks, and standardized evaluation protocols. While Table 1 provides comprehensive metrics across all baselines, the revised appendix will include detailed statistical analysis and simulation parameters to enhance reproducibility.\n\n## Theoretical Framework\n\nWhile fiber bundle formalism necessitates rigorous mathematics, we'll add this intuitive analogy with a supporting diagram: fiber bundle as multi-story building where base space (state space) is the floor plan, fibers are vertical spaces representing possible measurements, safety certificates are structural supports, and connection forms are elevators maintaining coherence between horizontal (state) and vertical (measurement) dimensions. The key insight is that traditional approaches treat measurement uncertainty as an external disturbance, whereas our approach incorporates it as an intrinsic geometric property of the system structure. We will add a visualization of the fiber bundle structure in the revised version, showing how trajectories in state space are associated with measurement uncertainties in the total space.\n\n## Computational Complexity & High-Dimensionality\n\nOur algorithm demonstrates excellent scalability with complexity divided into: fiber bundle connection calculation (O(n²), n=state dimension), mCBF evaluation (O(n+m), m=measurement dimension), and safety-constrained optimization (O(nd), d=policy parameters). The revised version will include a new section on computational complexity analysis with runtime measurements across system dimensions (n=6: 1.2ms, n=12: 3.8ms, n=24: 9.5ms, n=48: 22.7ms), plus IEEE 14-bus power grid validation tests using PYPOWER in the appendix, where our method achieved 97.2% constraint satisfaction under line thermal limits of 85% and voltage stability constraints of ±0.05 p.u., demonstrating effective application to practical higher-dimensional systems.\n\n## Physical Validation on real Franka Arm\n\nOur revision conducted validation on a 7-DOF Franka arm in the real world with joint velocity limits (±1.0 rad/s), acceleration constraints (±2.0 rad/s²), end-effector pose stability (±5° deviation), force limits (5N), and power consumption constraints (≤60W). We applied Gaussian noise, dropouts, and delays to IMU/force sensors. Results show 93.5% constraint satisfaction versus 81.2% for baseline methods, and trajectory error of 2.3cm compared to the baseline's 4.8cm. The revised paper will include detailed experimental setup photos, hardware specifications, and constraint visualization plots. We identified challenges, including joint friction, inconsistent sensor frequencies, and constraint priority conflicts. Our geometric framework naturally addresses these via dynamic constraint better priority adjustment based on measurement quality compared with other methods.\n\n## Measurement Quality Adaptation\n\nWe created controlled testing scenarios with noise levels (0-8%) and data dropout rates (0-25%) for the simulation Franka task. Results show safety boundary adaptations: high-quality regions (noise <2%, dropout <5%) at 3.5cm, medium uncertainty (noise 4-6%) at 5.7cm, high uncertainty (noise >7%, dropout >20%) at 8.2cm, with fallback to conservative behavior during sensor failure. This geometric adaptation (from Equation 13) reduced path length by 25.3% versus fixed-boundary methods while maintaining safety, confirming the fiber bundle framework's effectiveness for varying measurement quality. The revised version will add an adaptive safety boundary analysis figure showing boundary values as a continuous function of measurement quality with corresponding robot trajectories under different sensing conditions.\n\n## Appendix Accessibility\n\nThe complete PDF contains all appendices, including the proof of Theorem 3.1 through six steps: preliminary assumptions, perfect measurement invariance, uncertainty propagation, local certification, temporal correlation analysis, and global safety guarantees. We can provide a separate file if needed. \n\n## Conclusion\n\nWe believe these clarifications and additional visualizations in the revised version will improve readability while confirming the effectiveness and practicality of our method through real system validation.", "claims_and_evidence": "The claims made in the submission are supported by both theoretical analysis and experimental evidence. The authors claim that their geometric framework provides a unified approach to handling measurement uncertainty, system dynamics, and constraints, which is substantiated by the detailed mathematical formulation of the fiber bundle structure and its properties. The claim that their approach leads to improved learning efficiency and constraint satisfaction is supported by experimental results, though the paper would benefit from more detailed quantitative comparisons with baseline methods. The theoretical guarantees of learning convergence and constraint satisfaction are rigorously derived from the geometric properties of the bundle structure, providing a solid foundation for the practical implementation of the framework.", "methods_and_evaluation_criteria": "The proposed methods are mathematically sound and well-suited to the problem of learning dynamics under environmental constraints with uncertain measurements. The authors develop a comprehensive geometric framework based on fiber bundle theory, which provides a natural setting for handling measurement uncertainty. The evaluation criteria include both theoretical analysis of convergence and safety guarantees, as well as practical demonstrations of the framework's effectiveness in simulation environments. The paper would benefit from more explicit descriptions of the baseline methods used for comparison and clearer metrics for quantifying improvements in learning efficiency and constraint satisfaction.", "theoretical_claims": "The paper makes several significant theoretical claims, including the formulation of measurement-adapted Control Barrier Functions, the convergence of the learning dynamics, and probabilistic safety guarantees. I have examined the theoretical development in Sections 3 and 4, including the key Theorem 3.1 which provides probabilistic safety guarantees. The proof sketch outlines a three-step approach that appears sound, though the full proof is referenced to be in Appendix A which was not available for review. The theoretical framework is well-grounded in differential geometry and control theory, with clear connections to established concepts such as fiber bundles, connections, and barrier functions.", "experimental_designs_or_analyses": "The paper mentions \"extensive simulations\" that demonstrate significant improvements in learning efficiency and constraint satisfaction, but the details of these experiments are somewhat limited in the sections I was able to review. The experimental design appears to involve learning dynamics models under various measurement uncertainty conditions, with comparisons to traditional methods. However, without more specific information about the simulation environments, performance metrics, and statistical analyses, it is difficult to fully assess the validity of the experimental claims. The paper would benefit from more detailed descriptions of the experimental setup and comprehensive results.", "relation_to_broader_scientific_literature": "The paper effectively situates its contributions within the broader scientific literature on safety-critical control, geometric learning, and measurement-aware control. The authors provide a comprehensive review of related work in Section 2, acknowledging foundational contributions in differential geometry (Ehresmann, 1950; Kobayashi & Nomizu, 1996), control barrier functions (Ames et al., 2019), and geometric learning (Chen et al., 2018). They clearly articulate how their approach addresses limitations in existing methods, particularly in handling measurement uncertainty as an intrinsic geometric property rather than an external disturbance. The paper builds upon and extends several important lines of research in a coherent and well-motivated manner.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths:\n- The paper presents a novel and mathematically rigorous geometric framework that unifies measurements, constraints, and dynamics learning.\n- The theoretical guarantees of learning convergence and constraint satisfaction provide a solid foundation for the practical implementation of the framework.\n- The approach naturally adapts to local sensing conditions, making it particularly suitable for real-world applications with uncertain measurements.\n- The integration with Neural ODEs enables learning continuous-time dynamics while preserving geometric constraints.\n\nWeaknesses:\n- The paper is highly theoretical and may be challenging for readers without a strong background in differential geometry and control theory.\n- The experimental validation could be more comprehensive, with clearer comparisons to baseline methods and more detailed quantitative results.\n- The practical implementation details of the framework, particularly for complex systems, are somewhat limited.\n- The paper does not extensively discuss the computational complexity of the approach or potential scalability issues.", "other_comments_or_suggestions": "- The paper would benefit from more illustrative examples or case studies to demonstrate the practical application of the theoretical framework.\n- A more detailed discussion of the computational aspects of implementing the framework, particularly for high-dimensional systems, would strengthen the paper.\n- The authors could consider providing more intuitive explanations of the key geometric concepts to make the paper more accessible to a broader audience.\n- Additional visualizations of the fiber bundle structure and its relationship to measurement uncertainty would help readers better understand the geometric framework.", "questions_for_authors": "- How does the computational complexity of your approach scale with the dimensionality of the state space and the complexity of the constraints? Are there practical limitations to applying this framework to high-dimensional systems?\n- The paper focuses on theoretical guarantees and simulation results. Have you considered or tested the application of this framework on physical systems with real sensor measurements? What additional challenges might arise in such settings?\n- The measurement-adapted Control Barrier Functions (mCBFs) adapt to local sensing conditions. Could you elaborate on how this adaptation mechanism performs in environments with highly variable sensing quality, such as those with occlusions or sensor failures?", "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": {"ethical_review_concerns": "N/A"}}]

Author rebuttal:

Meta-review: This paper takes a geometric perspective on learning and control for systems with safety constraints. The reviewers appreciate the strength of this contribution, which is theoretically solid while also achieving impressive performance on numerical experiments. There is consensus among the two reviewers who engaged in discussions, while the third review is more surface level (and was flagged as likely AI generated).
Decision: Accept (spotlight poster)