Learning Dynamics under Environmental Constraints via Measurement-Induced Bundle Structures

Dongzhe Zheng; Wenjie Mei

REVIEW 4 major objections 5 minor 37 references

Learning Dynamics under Environmental Constraints via Measurement-Induced Bundle Structures

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that a robot's locally noisy sensor readings are not noise to be filtered out but a geometric structure — a fiber bundle over state space — that turns each measurement into an adaptive safety certificate and drives…

desk verdict Well-executed empirical paper whose two main theorems don't prove what the claims say — the convergence proof runs on an oracle update and the safety proof's final bound doesn't follow — so it's not citable as theory. read the letter →

arxiv 2505.19521 v2 pith:O5WJLAQC submitted 2025-05-26 cs.RO cs.LGcs.SYeess.SY

Dongzhe Zheng , Wenjie Mei This is my paper · ORCID

classification cs.ROcs.LGcs.SYeess.SY

keywords fiberbundlemeasurement-adaptedcontrolbarrierfunctionssafelearningdynamicsNeuralODEmeasurementuncertaintysafetycertificateslocalsensing

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to establish that a robot can learn unknown dynamics and guarantee safety using only local, noisy measurements, by treating measurement uncertainty as geometry rather than as an external disturbance. The central construction is a fiber bundle over the state space whose fiber above each state contains every measurement the sensors could return for that state; on this bundle the paper defines measurement-adapted control barrier functions, safety certificates whose margins tighten automatically where sensing is poor. With such a certificate, the paper proves that the probability of staying inside the safe set for all future time is at least $1 - \exp(-c/\delta_v^2)$, and that a Neural ODE trained within this geometry converges to the true dynamics exponentially fast, up to a residual proportional to the measurement noise bound $\delta_v$. If these claims are right, robots could learn and act safely with cheap, imperfect sensors and no global description of the constraints, and both safety and learning quality would degrade in a predictable way as sensing worsens.

What carries the argument

The load-bearing object is the measurement-induced fiber bundle $\pi: E = M \times Y \to M$ with fiber $\pi^{-1}(x) = \{(x, y) : y = h(x) + v,\ \|v\| \le \delta_v\}$, together with a connection $\nabla$ that couples state evolution with measurement evolution through a measurement-feedback gain operator $K$. On this bundle the paper defines the measurement-adapted control barrier function (mCBF) $b: E \to \mathbb{R}$, whose three defining conditions do the real work: the implication $b(x,y) \ge 0 \Rightarrow x \in S_0$ ties the certificate to the safe set; the Lie-derivative condition $\inf_u[L_f b + (L_g b)w + \alpha(b)] \ge 0$ provides forward invariance along the controlled flow; and the Lipschitz condition $|b(x,y_1) - b(x,y_2)| \le L_b d_Y(y_1, y_2)$ converts bounded measurement error into bounded erosion of the safety margin. The connection propagates that margin along trajectories, and the sub-Gaussian tail of the measurement noise turns the erosion into the probability bound $1 - \exp(-c/\delta_v^2)$. For learning, the mechanism is the bundle-valued update on the estimate $\hat f$ combined with the uncertainty-weighted data loss of Eq. (14), whose inverse-covariance weights $\Sigma_i^{-1}$ feed the measurement geometry directly into the gradient.

What would settle it

Train the implemented rule of Eq. (14) on any of the three tasks while measuring the bundle-norm error $\|\hat f - f\|_E$ against training time and noise bound $\delta_v$: the paper predicts decay like $c_1 e^{-\lambda_1 t} + c_2 \delta_v$, so a plateau well above $c_2 \delta_v$ or non-exponential decay would contradict that prediction for the implemented algorithm. Independently, count constraint violations over many seeds at several noise levels: the safety claim predicts a rate scaling like $\exp(-c/\delta_v^2)$, so a log-linear plot against $1/\delta_v^2$ that is markedly flatter than the predicted slope would contradict Theorem 3.1. The paper itself notes the implementation is limited for highly stochastic dynamics, the regime where the sub-Gaussian-noise assumption behind the exponential bound fails; a test with heavy-tailed noise would probe that boundary.

Watch

Extended reading notes

Core claim

The paper's claim is that the pairing of a state manifold with its uncertain measurements is itself a fiber bundle $\pi: E = M \times Y \to M$ with fiber $\pi^{-1}(x) = \{(x,y) : y = h(x) + v,\ \|v\| \le \delta_v\}$, and that this induced geometry carries the entire argument. A measurement-adapted control barrier function $b: E \to \mathbb{R}$ is a bundle map that certifies safety through $b(x,y) \ge 0 \Rightarrow x \in S_0$, satisfies the forward-invariance condition $\inf_{u \in U}[L_f b + (L_g b)w + \alpha(b)] \ge 0$ along admissible controls, and is $L_b$-Lipschitz in the measurement direction. Theorem 3.1 states that whenever such a certificate starts nonnegative, every admissible noise sequence satisfies $P(x(t) \in S_0 \text{ for all } t \ge 0) \ge 1 - \exp(-c/\delta_v^2)$. Theorem 4.1 states that an estimator trained through the bundle's uncertainty-weighted loss satisfies $\|\hat f - f\|_E \le c_1 e^{-\lambda_1 t} + c_2 \delta_v$: the learned model converges exponentially to the true dynamics with a noise-limited floor, while the safety bound continues to hold with a constant $c_3$. Together these assert that sensing quality sets the achievable rate of both safety and learning.

Load-bearing premise

The learning-convergence guarantee of Theorem 4.1 is proved for the update law in Eq. (12), whose right-hand side contains the true, unknown dynamics $f$, while the training rule actually run in the experiments, Eq. (14), learns from data that never reveal $f$; unless the two rules are shown to behave identically, the exponential convergence bound belongs to the proof's idealized process rather than to the implemented algorithm.

Editorial extensions

If this is right

Safety becomes location-adaptive: where sensors are accurate the certificate relaxes, where they are uncertain it tightens, and a single certificate works without a global map of the obstacle field.
The probability of ever leaving the safe set is bounded by $\exp(-c/\delta_v^2)$, so the worst-case violation rate decays faster than any polynomial as sensing sharpens.
The learned model approaches the true dynamics in the bundle norm at an exponential rate, with a floor $c_2 \delta_v$ set by measurement noise; learning quality is therefore limited by sensing quality rather than by the neural architecture.
Because no global constraint knowledge is assumed, the same certificate transfers to new obstacle configurations, which the paper supports with a reported 96.3% success rate across the worm, arm, and quadrotor tasks.
Policy updates can be projected onto the certificate's safe-policy set, which the paper offers as a route for reinforcement learning with partial, uncertain observations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One consequence the authors leave implicit is a sensor-economics trade: any change to the measurement map or to sensor placement that shrinks the product $L_b \delta_v$ directly tightens the safety bound and lowers the learning floor, so choosing sensors becomes an optimization on the bundle geometry rather than a filtering problem.
The safety theorem is stated for certificates built on the true dynamics $f$; extending it to a controller that runs on the learned $\hat f$ requires showing the mCBF conditions survive the estimation error, for instance by feeding Theorem 4.1's bound back into the invariance condition, a loop the paper does not close.
The bound's dependence on sensing quality is sharp enough to test: plotting constraint-violation frequency against $1/\delta_v^2$ on a log scale in the three simulated tasks should show the predicted $\exp(-c/\delta_v^2)$ decay, a stronger fingerprint than a generic 'more noise, more violations' trend.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a serious empirical paper wearing a theory coat that doesn't fit. The three simulation tasks are sensible, the ablations are thorough, and the reported gains over baselines are plausible engineering results. But the theoretical guarantees, which are the main selling point, fail on close reading.

Theorem 3.1's proof in Appendix A ends with a geometric series that evaluates to exp(c1 Δt − c2/δ_v^2), which is ≥1 under the paper's own condition c1 Δt ≥ c2/δ_v^2. The bound doesn't follow. Lemma A.4 also just assumes temporal decorrelation without an argument. Theorem 4.1 is worse: the convergence proof differentiates a Lyapunov function along Eq. (12), dot(hat f) = −L1(hat f − f). That update law has the true dynamics f on the right-hand side. The implemented rule, Eq. (14), is gradient descent on a data loss and never sees f. So the exponential convergence is proven for an oracle process, not for the proposed algorithm. The extra δ_v term in Eq. (38) also appears from nowhere. These are load-bearing, not cosmetic.

What's new? The mCBF conditions reduce essentially to a robust CBF with a Lipschitz measurement margin; Appendix G spells out b(x,y)=b0(x,h(x))−L_b||y−h(x)||, which is close to Taylor & Ames's adaptive safety. The fiber-bundle language doesn't add mathematical content that I can see. The experiments do show consistent improvements, and the code link and hyperparameter detail help reproducibility, though there is no commit hash. The noise model is also internally inconsistent: y=h(x)+v with |v|≤δ_v, but the proof uses a sub-Gaussian tail bound.

Bottom line: the empirical part is worth a look for people building measurement-aware safety filters, but the theoretical claims should not be relied on. I'd send it to reviewers only to confirm the proof failures; as a contribution it needs major restructuring. If the authors decouple the empirical recipe from the false guarantees and either fix or drop the theorems, there might be a usable paper.

Referee Report

4 major / 5 minor

Summary. The paper proposes a geometric framework in which measurement uncertainty induces a fiber bundle over the state space, used to define measurement-adapted control barrier functions (mCBFs) and a Neural ODE-based dynamics learner. The main theoretical claims are Theorem 3.1, a probabilistic safety guarantee of the form P(x(t) in S0 for all t >= 0) >= 1 - exp(-c/delta_v^2), and Theorem 4.1, an exponential convergence bound for the learned dynamics with a measurement-noise residual together with a safety guarantee during learning. The paper also presents simulation results on soft-body, manipulator, and quadrotor tasks, plus ablations and cross-domain experiments.

Significance. If established, the framework would be a noteworthy unification of geometric control and learning: the safety guarantee degrades gracefully with measurement noise, and the convergence bound is parameterized by sensing quality. The paper ships a public implementation and reports extensive experiments and ablations, which is a strength. However, the theoretical results are the main advertised contribution, and the proofs contain load-bearing gaps: the safety proof's final union bound evaluates to a trivial bound, and the convergence proof analyzes an oracle update that uses the unknown true dynamics rather than the implemented parameter update. Until these issues are resolved, the theoretical significance claimed in the paper is not supported.

major comments (4)

[Appendix A, Eq. (35)] The union bound is evaluated incorrectly. The series sum_{k=0}^infty exp(-c2/delta_v^2) (1 - exp(-c1 Delta t))^k equals exp(c1 Delta t - c2/delta_v^2), which is at least 1 under the stated condition c1 Delta t >= c2/delta_v^2. Therefore Eq. (35) does not imply the claimed probability bound, and Theorem 3.1 is unproven. In addition, Lemma A.4 is asserted with no derivation; it requires a temporal decorrelation property of the events A_t, but these events depend on the state trajectory, so independence or exponential mixing is not automatic from sub-Gaussian measurement noise.
[Section 4.1-4.3, Eqs. (12) and (14)] Theorem 4.1's convergence analysis is performed for the update dot(hat f) = -L1(hat f - f), whose right-hand side contains the unknown true dynamics f. The implemented rule in Eq. (14) updates parameters theta by gradient descent on a data loss and is never shown to realize Eq. (12); no relation between theta and hat f is given, and no bound on dot(hat f_theta) + L1(hat f_theta - f) is supplied. The exponential convergence bound therefore applies to an oracle process, not to the proposed algorithm.
[Appendix B, Eq. (38)] The Lyapunov derivative contains the term ||hat f - f||_E delta_v, but from dot(hat f) = -L1(hat f - f) one obtains dot(V) = -<hat f - f, L1(hat f - f)> exactly. The measurement-noise residual c2 delta_v is inserted rather than derived, so even the oracle convergence bound with a noise-dependent residual is not established.
[Appendix B, Eq. (44)] The proof of the second inequality in Theorem 4.1 assumes b(x(t), h(x(t))) >= b_min for all t along the learning rollout. Lemma A.1 establishes a similar statement only under perfect measurements (v = 0) and without the policy/learning coupling; the effect of the learned dynamics and the actual measurement noise on the mCBF value is not analyzed. Consequently, the safety guarantee during learning, P(x(t) in S0) >= 1 - exp(-c3/delta_v^2), is not proven, and the statement's quantifier over t is ambiguous.

minor comments (5)

[Section 3.2, Eq. (3)] The connection formula uses pi_*^{-1} and the term K(x)(y - h(x)); these are not defined as vector fields on E, so the expression is not mathematically well-formed as written.
[Section 3.6, Eq. (9)] The mCBF condition omits explicit dependence on the control input u in L_f b; the infimum over u is taken, but f(x,u) and the chosen control law should be specified.
[Section 5.2] The sensor noise model ||v_i|| <= alpha ||x - x_i|| is state-dependent, while the theory assumes a global bound ||v|| <= delta_v; the relation between the two should be stated.
[Appendix D] The additional environments are described as motivated by real datasets but are themselves simulations; the text should clarify that no real sensor data are used.
[Theorems 3.1 and 4.1] The theorems state constants c, c1, c2, c3, lambda1 without specifying their dependence on system parameters; the proofs should state the relevant dependencies for the claims to be checkable.

Circularity Check

1 steps flagged · score 8.0 of 10

Theorem 4.1's exponential convergence is proven for an oracle update law (Eq. 12) that contains the unknown true dynamics f, not for the implemented gradient rule (Eq. 14), making the central learning guarantee circular by construction.

self definitional [Section 4.1, Eqs. (12) and (14); Theorem 4.1; Appendix B]
"The learning dynamics on the bundle take the form ˙ˆf=−L 1( ˆf−f) ... (12) ... The learning process incorporates measurement uncertainty through: ˙θ=−Λ∇ θT( ˆfθ,D) (14) ... Theorem 4.1. Under the proposed learning dynamics, we have ∥ ˆf−f∥ E ≤c 1 exp(−λ1t) +c 2δv"

The proof of Theorem 4.1 in Appendix B differentiates V(t)=1/2||fhat−f||_E^2 along dot(fhat)=−L1(fhat−f). Because f appears on the right-hand side of this update law, dot(fhat) is driven directly toward the unknown true dynamics; for any positive-definite L1 the exponential decay is true by construction. But the implemented rule (14) updates parameters θ by gradient descent on the data loss T(fhat_θ,D)=Σ||fhat(x_i,u_i)−dot(x)_i||^2 and never observes f. The paper supplies no map from θ to fhat under which (14) equals (12), and no bound on the error introduced by replacing (12) with (14).

full rationale

The paper's central learning claim (first inequality of Theorem 4.1) is established in Appendix B by differentiating V = 1/2||fhat−f||_E^2 along dot(fhat) = −L1(fhat−f). That ODE contains the true f as a target signal, so convergence of fhat to f is true by construction. The algorithm actually implemented is Eq. (14), dot(theta) = −Lambda grad_theta T(fhat_theta,D), which is data-driven gradient descent and never accesses f. No relationship is given under which (14) realizes (12), nor is the mismatch dot(fhat_theta) + L1(fhat_theta − f) bounded. Hence the theorem describes an oracle process; the prediction 'learning convergence' is not derived for the proposed method. The second inequality of Theorem 4.1 repeats Theorem 3.1's safety claim and inherits its issues. Appendix A's union-bound series actually evaluates to exp(c1 Delta t − c2/delta_v^2), which is at least 1 under the stated condition c1 Delta t >= c2/delta_v^2, rather than < exp(−c/delta_v^2); this is a correctness failure rather than circularity. Appendix B also inserts the +||fhat−f||_E delta_v term in (38) with no source in the oracle ODE, so the c2 delta_v residual is asserted, not derived. No load-bearing self-citation chain appears; the circularity is definitional and concentrated in the convergence theorem, which is the paper's main theoretical guarantee. Simulations are external and do not cure the oracle gap. Overall score 8.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

All claims rest on the existence of an mCBF satisfying (9), a known measurement map h and noise bounds, an oracle learning update in Eq (12), and an unstated noise-mixing assumption in Lemma A.4. The nominal safety certificate b0, class-K functions alpha and beta, the Lipschitz constant L_b, the operator L1, and the connection gain K are design choices with no construction given. No new physical entities are postulated.

free parameters (5)

L_b (measurement Lipschitz constant of b)
Appears in the safety margin b(x,h(x))-L_b||y-h(x)|| and in c3=b_min^2/(2 L_b^2); no principled construction or fitting procedure is given.
b0 / Phi0 nominal safety certificate
The safety guarantee assumes a nominal certificate b0 beneath the mCBF; existence across the state space is not established.
alpha, beta class K functions
These functions define how safety margins shrink with measurement error (Eqs 13 and 47); their choice is left to the user.
lambda_min(L1)
Convergence rate in Theorem 4.1 is the minimum eigenvalue of an operator L1 whose construction is not specified.
measurement feedback gain K(x)
Defines the bundle connection in Eq (3) but is never used in the proofs; it is effectively an unused design choice.

assumptions (5)

domain assumption Known smooth measurement map h and known noise bounds delta_w, delta_v on process and measurement noise.
Section 3.1 defines y=h(x)+v with |v|<=delta_v and |w|<=delta_w; the mCBF inequalities and the safety probability all depend on these bounds being available.
ad hoc to paper There exists an mCBF b satisfying conditions (9): b>=0 implies x in S0, the Lie-derivative inequality for all admissible w, and global Lipschitz continuity in y.
No construction is given; Appendix G only illustrates b=b0(x,h(x))-L_b||y-h(x)|| for a special case. The main theorem assumes the object whose existence is the practical difficulty.
ad hoc to paper The learning update in Eq (12) has access to the true dynamics f.
dot(hat f)=-L1(hat f-f); this oracle premise is required by Appendix B's Lyapunov proof, but the implemented optimizer (14) does not have f.
domain assumption Measurement noise is sub-Gaussian and temporally decorrelates fast enough for Lemma A.4's product bound.
Lemma A.4 assumes P(At and At') <= exp(-min{c1|t-t'|, c2/delta_v^2}); no mixing assumption is stated, and Section 3.1's bounded-noise statement conflicts with the Gaussian tail.
standard math Gronwall's inequality, the comparison lemma, and standard properties of class K and K-infinity functions.
Used in Lemma A.1, Lemma A.2, and Appendix B without proof; these are standard background tools.

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Cite this review

Pith. "Pith review of Learning Dynamics under Environmental Constraints via Measurement-Induced Bundle Structures." pith.science (2026). https://pith.science/paper/O5WJLAQC

@misc{pith2026250519521,
  author       = {Pith},
  title        = {Pith review of: Learning Dynamics under Environmental Constraints via Measurement-Induced Bundle Structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O5WJLAQC}},
  note         = {Machine review of arXiv:2505.19521}
}

read the original 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.

Figures

Figures reproduced from arXiv: 2505.19521 by the authors.

**Figure 1.** Illustration of three experimental tasks from left to right: A soft-bodied worm robot navigating through obstacles using peristaltic motion, a 7-DOF Franka robotic arm performing obstacleaware joint motion, and a 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. For al… view at source ↗

**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 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Average success rates across three tasks under different noise levels. Comparison between our method, Neural-CBF, SafetyNet, GPMPC and DataFilter. Error bars indicate standard deviation across 10 runs. adapt under environmental constraints through local observations, bridging the gap between theoretical control guarantees and modern robotics (or even practical embodied intelligence). The measurement-induced bundl… view at source ↗

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Works this paper leans on

37 extracted references · 28 canonical work pages

[1]

D., Xu, X., Grizzle, J

Ames, A. D., Xu, X., Grizzle, J. W., and Tabuada, P. Control barrier function based quadratic programs for safety critical systems. IEEE Transactions on Automatic Control, 62 0 (8): 0 3861--3876, 2016

2016
[2]

D., Coogan, S., Egerstedt, M., Notomista, G., Sreenath, K., and Tabuada, P

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. IEEE, 2019

2019
[3]

Safe model-based reinforcement learning with stability guarantees

Berkenkamp, F., Turchetta, M., Schoellig, A., and Krause, A. Safe model-based reinforcement learning with stability guarantees. Advances in Neural Information Processing Systems, 30, 2017

work page 2017
[4]

Berkenkamp, F., Krause, A., and Schoellig, A. P. Bayesian optimization with safety constraints: safe and automatic parameter tuning in robotics. Machine Learning, 112 0 (10): 0 3713--3747, 2023

2023
[5]

D., Paulson, J

Bonzanini, A. D., Paulson, J. A., Makrygiorgos, G., and Mesbah, A. Fast approximate learning-based multistage nonlinear model predictive control using gaussian processes and deep neural networks. Computers & Chemical Engineering, 145: 0 107174, 2021

work page 2021
[6]

An introduction to optimization on smooth manifolds

Boumal, N. An introduction to optimization on smooth manifolds. Cambridge University Press, 2023

work page 2023
[7]

M., Bruna, J., LeCun, Y., Szlam, A., and Vandergheynst, P

Bronstein, M. M., Bruna, J., LeCun, Y., Szlam, A., and Vandergheynst, P. Geometric deep learning: going beyond euclidean data. IEEE Signal Processing Magazine, 34 0 (4): 0 18--42, 2017

2017
[8]

T., Rubanova, Y., Bettencourt, J., and Duvenaud, D

Chen, R. T., Rubanova, Y., Bettencourt, J., and Duvenaud, D. K. Neural ordinary differential equations. Advances in Neural Information Processing Systems, 31, 2018

work page 2018

Show all 37 references

[9]

M., and Burdick, J

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 Intelligence, volume 33, pp.\ 3387--3395, 2019

2019
[10]

J., Lee, D., Sreenath, K., Tomlin, C

Choi, J. J., Lee, D., Sreenath, K., Tomlin, C. J., and Herbert, S. L. Robust control barrier--value functions for safety-critical control. In 2021 60th IEEE Conference on Decision and Control (CDC), pp.\ 6814--6821. IEEE, 2021

2021
[11]

Gauge equivariant convolutional networks and the icosahedral cnn

Cohen, T., Weiler, M., Kicanaoglu, B., and Welling, M. Gauge equivariant convolutional networks and the icosahedral cnn. In International Conference on Machine Learning, pp.\ 1321--1330. PMLR, 2019

2019
[12]

and Murray, R

Da s , E. and Murray, R. M. Robust safe control synthesis with disturbance observer-based control barrier functions. In 2022 IEEE 61st Conference on Decision and Control (CDC), pp.\ 5566--5573. IEEE, 2022

2022
[13]

R., and Elworthy, D

Diaconis, P., Elworthy, D., F \"o llmer, H., Nelson, E., Papanicolaou, G., Varadhan, S. R., and Elworthy, D. Geometric aspects of diffusions on manifolds. In \'E cole d' \'E t \'e de Probabilit \'e s de Saint-Flour XV--XVII, 1985--87 , pp.\ 277--425. Springer, 1988

1985
[14]

Neural networks with physics-informed architectures and constraints for dynamical systems modeling

Djeumou, F., Neary, C., Goubault, E., Putot, S., and Topcu, U. Neural networks with physics-informed architectures and constraints for dynamical systems modeling. In Learning for Dynamics and Control Conference, pp.\ 263--277. PMLR, 2022

2022
[15]

Les connexions infinit \'e simales dans un espace fibr \'e diff \'e rentiable

Ehresmann, C. Les connexions infinit \'e simales dans un espace fibr \'e diff \'e rentiable. In Colloque de Topologie, Bruxelles, volume 29, pp.\ 55--75, 1950

1950
[16]

Gurriet, T., Mote, M., Singletary, A., Nilsson, P., Feron, E., and Ames, A. D. A scalable safety critical control framework for nonlinear systems. IEEE Access, 8: 0 187249--187275, 2020

2020
[17]

Gem: Group enhanced model for learning dynamical control systems

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, 2021

2021 arXiv
[18]

Hsu, E. P. Stochastic analysis on manifolds. Number 38. American Mathematical Soc., 2002

2002
[19]

Robust control barrier functions for constrained stabilization of nonlinear systems

Jankovic, M. Robust control barrier functions for constrained stabilization of nonlinear systems. Automatica, 96: 0 359--367, 2018

2018
[20]

Kalman, R. E. A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82 0 (1): 0 35--45, 1960

1960
[21]

and Nomizu, K

Kobayashi, S. and Nomizu, K. Foundations of differential geometry, volume 2, volume 61. John Wiley & Sons, 1996

1996
[22]

Crafting papers on machine learning

Langley, P. Crafting papers on machine learning. In Langley, P. (ed.), Proceedings of the 17th International Conference on Machine Learning (ICML 2000), pp.\ 1207--1216, Stanford, CA, 2000. Morgan Kaufmann

2000
[23]

Lewis, A. D. Affine connections and distributions with applications to nonholonomic mechanics. Reports on Mathematical Physics, 42 0 (1): 0 135--164, 1998. Proceedings of the Pacific Institute of Mathematical Sciences Workshop on Nonholonomic Constraints in Dynamics

1998
[24]

System-level safety guard: Safe tracking control through uncertain neural network dynamics models

Li, X., Li, Y., Girard, A., and Kolmanovsky, I. System-level safety guard: Safe tracking control through uncertain neural network dynamics models. In 6th Annual Learning for Dynamics & Control Conference, pp.\ 968--979. PMLR, 2024

2024
[25]

Safe control under input limits with neural control barrier functions

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

1970
[26]

and Weinstein, A

Marsden, J. and Weinstein, A. Reduction of symplectic manifolds with symmetry. Reports on Mathematical Physics, 5 0 (1): 0 121--130, 1974

1974
[27]

Gauge theory of the falling cat

Montgomery, R. Gauge theory of the falling cat. Fields Institute Communications, 1 0 (10.1090), 1993

1993
[28]

D., Issac, J., Kappler, D., Birchfield, S., and Fox, D

Ratliff, N. D., Issac, J., Kappler, D., Birchfield, S., and Fox, D. Riemannian motion policies. arXiv preprint arXiv:1801.02854, 2018

2018 arXiv
[29]

Active learning of discrete-time dynamics for uncertainty-aware model predictive control

Saviolo, A., Frey, J., Rathod, A., Diehl, M., and Loianno, G. Active learning of discrete-time dynamics for uncertainty-aware model predictive control. IEEE Transactions on Robotics, 2023

2023
[30]

Learning for safety-critical control with control barrier functions

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

2020
[31]

Taylor, A. J. and Ames, A. D. Adaptive safety with control barrier functions. In 2020 American Control Conference (ACC), pp.\ 1399--1405. IEEE, 2020

2020
[32]

Safetynet: Safe planning for real-world self-driving vehicles using machine-learned policies

Vitelli, M., Chang, Y., Ye, Y., Ferreira, A., Wo czyk, M., Osi \'n ski, 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 Au...

2022
[33]

P., Taylor, A

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-jacobi reachability, control barrier functions, and predictive methods for uncertain systems. IEEE Control Systems Magazine, 43 0 (...

2023
[34]

Structure-preserving algorithms for oscillatory differential equations II

Wu, X., Liu, K., and Shi, W. Structure-preserving algorithms for oscillatory differential equations II. Springer, 2015

2015
[35]

Towards generalist robots: A promising paradigm via generative simulation

Xian, Z., Gervet, T., Xu, Z., Qiao, Y.-L., Wang, T.-H., and Wang, Y. Towards generalist robots: A promising paradigm via generative simulation. arXiv preprint arXiv:2305.10455, 2023

2023 arXiv
[36]

Geometric reinforcement learning for path planning of uavs

Zhang, B., Mao, Z., Liu, W., and Liu, J. Geometric reinforcement learning for path planning of uavs. Journal of Intelligent & Robotic Systems, 77: 0 391--409, 2015

2015
[37]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

D., Xu, X., Grizzle, J

Ames, A. D., Xu, X., Grizzle, J. W., and Tabuada, P. Control barrier function based quadratic programs for safety critical systems. IEEE Transactions on Automatic Control, 62 0 (8): 0 3861--3876, 2016

2016

[2] [2]

D., Coogan, S., Egerstedt, M., Notomista, G., Sreenath, K., and Tabuada, P

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. IEEE, 2019

2019

[3] [3]

Safe model-based reinforcement learning with stability guarantees

Berkenkamp, F., Turchetta, M., Schoellig, A., and Krause, A. Safe model-based reinforcement learning with stability guarantees. Advances in Neural Information Processing Systems, 30, 2017

work page 2017

[4] [4]

Berkenkamp, F., Krause, A., and Schoellig, A. P. Bayesian optimization with safety constraints: safe and automatic parameter tuning in robotics. Machine Learning, 112 0 (10): 0 3713--3747, 2023

2023

[5] [5]

D., Paulson, J

Bonzanini, A. D., Paulson, J. A., Makrygiorgos, G., and Mesbah, A. Fast approximate learning-based multistage nonlinear model predictive control using gaussian processes and deep neural networks. Computers & Chemical Engineering, 145: 0 107174, 2021

work page 2021

[6] [6]

An introduction to optimization on smooth manifolds

Boumal, N. An introduction to optimization on smooth manifolds. Cambridge University Press, 2023

work page 2023

[7] [7]

M., Bruna, J., LeCun, Y., Szlam, A., and Vandergheynst, P

Bronstein, M. M., Bruna, J., LeCun, Y., Szlam, A., and Vandergheynst, P. Geometric deep learning: going beyond euclidean data. IEEE Signal Processing Magazine, 34 0 (4): 0 18--42, 2017

2017

[8] [8]

T., Rubanova, Y., Bettencourt, J., and Duvenaud, D

Chen, R. T., Rubanova, Y., Bettencourt, J., and Duvenaud, D. K. Neural ordinary differential equations. Advances in Neural Information Processing Systems, 31, 2018

work page 2018