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Extended Neural Contractive Dynamical Systems: On Multiple Tasks and Riemannian Safety Regions

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that a neural network whose Jacobian is always negative definite yields robot motion skills that are contractive by construction, and that this same architecture can handle multiple tasks and latent-space obstacle…

desk verdict The paper's central contraction guarantee fails: the line integral in Eq. (4) does not produce a negative definite Jacobian for arbitrary network outputs, so the core stability claim and everything built on it does not hold as stated. read the letter →

arxiv 2411.11405 v3 pith:QYSMHLQ6 submitted 2024-11-18 cs.RO cs.LG

Hadi Beik Mohammadi , S{o}ren Hauberg , Georgios Arvanitidis , Gerhard Neumann , Leonel Rozo This is my paper · ORCID
classification cs.ROcs.LG
keywords neuralcontractivedynamicalsystemscontractiontheorylearningfromdemonstrationobstacleavoidanceRiemannianmanifoldsvariationalautoencoderslatentspacemulti-taskrobot
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 argues that stable robot motion can be learned from demonstrations without giving up expressiveness, by building stability into the network architecture itself rather than adding constraints during training. The core construction defines the network's Jacobian as a negative-definite matrix for every parameter setting, then obtains the velocity field by integrating that Jacobian, so contraction—the exponential convergence of nearby trajectories to one another—holds for any learned parameters. The paper extends this idea in three directions: a learned contraction-spread regularizer that improves convergence toward demonstrated data, conditioning on task variables so one network can reproduce multiple skills, and latent-space obstacle avoidance that uses the Riemannian pullback metric of a variational autoencoder to mark obstacles and out-of-distribution regions as unsafe. If correct, the result is a practical recipe for learning-from-demonstration systems that are both flexible and certified stable, including in high-dimensional robot and human motion.

What carries the argument

The load-bearing object is the constructed Jacobian $\hat J_f(x)=-(J_\theta(x)^{\mathsf T}J_\theta(x)+\operatorname{diag}(\epsilon))$, a matrix that is negative definite for every network output because it is a negative Gramian minus a positive diagonal shift. The velocity field is the line integral of this Jacobian, so the fundamental theorem of line integrals guarantees that the Jacobian of the resulting field is exactly the constructed one, making contraction a property of the architecture rather than a property of training. Around this core, the paper adds a learned contraction-spread regularizer, a conditioning variable concatenated to the state, and a Riemannian modulation matrix $G_M(z)$ built from the gradient of a distance field rescaled by the inverse volume $\alpha/\sqrt{\det M(z)}$ of the pullback metric; this matrix reshapes the latent field near obstacles and low-density regions.

What would settle it

Compute the symmetric part of the Jacobian of the modulated latent field $\hat f(z)=G_M(z)f(z)$ on a dense grid around a concave obstacle: if any eigenvalue is positive, or if two nearby integral curves starting in the same connected component of the safe set ever separate, the contraction-preservation claim is false.

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Extended reading notes

Core claim

The central claim is that an NCDS—a neural network whose Jacobian is parameterized as $\hat J_f(x)=-(J_\theta(x)^{\mathsf T}J_\theta(x)+\operatorname{diag}(\epsilon))$ and whose output is the line integral of this Jacobian—is contractive for all parameters, so no post-hoc stabilization or constrained optimization is needed. The paper further claims that a state-independent regularization vector, which encourages spread between the Jacobian's eigenvalues, yields faster convergence toward the demonstrated region and better generalization; that conditioning the network on an extra input variable allows a single model to represent multiple motion skills; and that modulating the latent vector field with a matrix built from a distance field rescaled by the inverse pullback metric volume lets the robot avoid obstacles and out-of-data-support regions while retaining contraction. The experiments are presented as showing that only this method among the tested baselines produces monotonically decreasing distances between nearby trajectories, and that the latent structure lets it scale to 8- and 44-dimensional state spaces.

Load-bearing premise

The entire obstacle-avoidance and safety guarantee rests on the assumption that the matrix that reshapes the latent vector field near obstacles preserves contraction and does not pull trajectories into new false targets, and the paper adopts this assumption from earlier work instead of proving it for its learned Riemannian pullback setting.

Editorial extensions

If this is right

  • Learned robot skills can be deployed with a certified exponential-convergence guarantee to a target, without solving an optimization problem at execution time.
  • A single trained model can switch between different motion skills by changing a conditioning input such as a target state, trajectory shape, or image embedding, which reduces the need to train separate models.
  • Obstacle avoidance can be computed in a low-dimensional latent space instead of the high-dimensional ambient space, which lowers computational cost and makes stabilization practical for higher-dimensional robot and human motion.
  • Contraction guarantees extend to full-pose motions on $SO(3)$ or quaternions by restricting the decoder's output to the first cover of the Lie algebra, so orientation dynamics remain stable.
  • The approach gives a concrete alternative to asymptotic-stability methods: because contraction controls all pairs of trajectories, the system remains stable under perturbations and extrapolation beyond the demonstrations.

Reading between the lines

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

  • The state-independent regularization result suggests that contraction-speed tuning can be separated from reconstruction loss; a curriculum or schedule on contraction spread might improve data efficiency, which the paper does not test.
  • The finding that a symmetric Jacobian suffices despite the theoretical expressivity of asymmetric Jacobians may be task-dependent; motions with strong rotational or curl components are a natural place to test whether the skew-symmetric term matters.
  • The Gaussian-shaped obstacle model used in the ambient metric limits the approach to smooth obstacle representations; sharp or highly nonconvex obstacles would likely require recomputing the distance field at query time, and the manual alpha-scaling procedure would need to become adaptive.
  • If contraction in the latent space is preserved by the injective decoder's pullback, the same guarantee could plausibly extend to other injective generative models such as normalizing flows, though the paper leaves that connection unexplored.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper extends the Neural Contractive Dynamical System (NCDS) architecture for learning contractive dynamical systems from demonstrations. The base construction defines the learned vector field as a straight-line line integral of a matrix-valued network that always outputs negative definite matrices, and the paper claims this yields a contractive vector field by construction. On top of this, the paper adds three regularization strategies, an asymmetric Jacobian variant, a conditional version (CNCDS) for multi-task learning, a latent-space formulation using VAEs with injective decoders, and a Riemannian-modulation mechanism for obstacle avoidance and safety regions. The experimental section evaluates the method on LASA trajectories, a 7-DoF Franka robot, human motion data, and a vision-conditioned grasp-and-drop task.

Significance. If the theoretical guarantee were valid, the paper would make a strong contribution: it would give a neural-network dynamical system with guaranteed contraction, multi-task conditioning, and latent-space obstacle avoidance, supported by extensive robot and human-motion experiments. The experimental effort is substantial, including real robot executions and a public benchmark. However, the central construction does not deliver the claimed guarantee: the line integral in Eq. (4) does not produce a vector field whose Jacobian is the negative definite network output, so the 'contractive by construction' claim is false. This flaw propagates to the CNCDS, latent NCDS, and Riemannian safety-region extensions, which all inherit the base contraction claim. The paper's main novelty and its stated stability guarantees therefore rest on an invalid derivation.

major comments (2)
  1. [Section 2.2, Eq. (4)] The construction does not produce a vector field with Jacobian Ĵ_f(x). Direct differentiation of Eq. (4) gives ∂f_i/∂x_j = ∫₀¹ [Ĵ_{ij}(c(t)) + t Σ_k ∂Ĵ_{ik}/∂c_j(c(t))(x_k - x0_k)] dt, where c(t)=x0+t(x-x0). The displacement-weighted second term is absent from Ĵ_f(x), and even the first term is an average over the segment rather than the value at x. The fundamental theorem of line integrals invoked after Eq. (4) applies only to exact 1-forms, i.e., only when ∂Ĵ_{ij}/∂x_k = ∂Ĵ_{ik}/∂x_j for all i,j,k; this integrability condition is not enforced by Eq. (3). Hence the Jacobian of f is not guaranteed to be negative definite. For a concrete counterexample within the paper's parameterization, take x0=0, ε<1, and Ĵ_f(x) = -2I + ε[[0, sin x2],[sin x2,0]], which is of the form (3) for Jθ the smooth square root of (2-ε)I - ε[[0, sin x2],[sin x2,0]]. Eq. (4) then yields f(x)=(-2x1 + ε(1-cos x2), -2x2 + εx1(1-cos x2)/x2) (continuous extension at x2=0), whose Jacobian at (L,0) is diag(-2, -2 + εL/2), which has a positive eigenvalue for L>4/ε. Thus the claim 'contractive by construction' is false, and the extensions in Secs. 2.3, 3.2, and 3.4 all inherit this invalid foundation.
  2. [Section 3.4, Eqs. (32)-(34)] The latent modulation matrix G_M(z) is asserted to preserve contraction and avoid spurious attractors, but no proof is given for the learned pullback-metric setting. The argument simply refers to Huber et al. (2019, 2022) and Koptev (2023), yet the conditions of those proofs are not verified for the distance field S_scaled(z)=α/V(z)S(z). In particular, the safe set induced by the metric volume may not be simply connected, and the gradient of S_scaled may not satisfy the regularity assumptions needed for the modulation theorem. The auxiliary tangential term β(x)G_M(z)g(z) in Eq. (34) is introduced heuristically, and its effect on the contraction properties of the modulated field is not analyzed. Consequently, the obstacle-avoidance and safety-region guarantees claimed in Sec. 4.6 are unsupported.
minor comments (4)
  1. [Section 2.2.1, Eq. (7)] The regularization loss L_ϵ spreads eigenvalues only relative to ϵ_1 rather than over all pairs; this arbitrary choice should be justified, since it may bias the learned contraction spread. In addition, the '276% faster convergence' claim is based on the convex-hull occupancy metric, which depends on an experimentally chosen hull margin; reporting pointwise Jacobian eigenvalue metrics would provide a more direct check.
  2. [Appendix 6.2] The scaling factor α is selected experimentally, and it directly controls the alignment of the computed obstacle boundary with the true boundary in Fig. 13. Since this parameter is hand-tuned, the claim of correct boundary alignment is not automatic and should be accompanied by a sensitivity analysis or an automated selection procedure.
  3. [Section 2.3] In CNCDS, the conditioning variable ϖ is concatenated to the state, but no dynamics are defined for ϖ; it is effectively a constant parameter. The negative definiteness of the augmented matrix in Eq. (11) does not by itself imply contraction of the x-subsystem unless the exact Jacobian property holds, which is already invalid per the first major comment. This point should be clarified.
  4. [Section 4.7] The ablation on activation functions shows that Sigmoid 'fails to reach and stop at its target,' which suggests a violation of the claimed contraction/attraction property. This is worth discussing in light of the theoretical guarantee, since the construction is supposed to be parameter-independent.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the contractive-stability claim is an architectural construction, and the main comparisons use external baselines; the self-citations are historical rather than load-bearing.

full rationale

The central NCDS claim is not circular. The paper defines the learned Jacobian as the negative definite matrix -(J_theta^T J_theta + eps I) in Eq. (3) and defines f by the line integral in Eq. (4); the contractivity assertion rests on the fundamental theorem of line integrals and Definition 1, not on any fitted value or on a measured quantity being fed back as a prediction. Whether the line integral in Eq. (4) really has Jacobian equal to the integrand is a mathematical correctness question (the matrix field must satisfy integrability conditions), but a false or unproved step of that kind is not a circularity under the requested definitions. The Riemannian obstacle-avoidance and safety-region extension imports contraction preservation from Huber et al. (2019, 2022) and Koptev (2023); these are external sources, and any failure of those arguments to carry over to the learned pullback metric is an unverified generalization gap, not a self-referential reduction. Self-citations to Beik-Mohammadi et al. (2021, 2023, 2024) are historical and the paper re-derives the NCDS construction in Section 2.2; performance is measured against ELCD, Euclideanizing flow, Imitation flow, SEDS, and public LASA and KIT datasets. The regularization section directly maximizes eigenvalue spread, so the higher spread reported in Table 1 is expected from the loss, but the paper does not rest its central claim on that table and uses an independent time-in-region measure for convergence. Overall, there is at most minor self-referential reporting, no load-bearing circularity.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The central line-integral construction is self-contained, but the extended safety-guarantee relies on imported contraction-invariance and modulation results, plus hand-set scalars and an experimentally chosen alpha. The paper's own new ingredients are the learned regularization vector, the conditional variable, and the pullback-metric modulation, all of which introduce tunable constants or assumptions.

free parameters (6)
  • Learned anchor state x0 and initial velocity x_dot0
    Optimized jointly with network parameters in Eq. 4; the line integral depends on the chosen anchor and initial velocity.
  • Regularization vector epsilon
    Learned in the state-independent and state-dependent variants (Eq. 6) to increase contraction spread; reparameterized with a small constant floor.
  • beta weight in regularization loss
    Scales L_epsilon in Eq. 7 and Eq. 8; no value is reported in the experiments.
  • alpha scaling factor
    Selected experimentally in Appendix 6.2 to align the approximate obstacle boundary with the real boundary; directly shapes the safety regions.
  • Obstacle Gaussian parameters o, r, w
    Hand-set in Eq. 26 to represent obstacle position, radius, and cost weight; these define the ambient metric that is pulled back to latent space.
  • Modulation sigmoid parameters rho, nu, k = rho=1, nu=10, k=2
    Configured in Eq. 37 following Koptev (2023); these control when modulation activates and how strongly the obstacle repels.
assumptions (5)
  • standard math The line integral of a C^1 matrix field along straight paths is differentiable and its Jacobian equals the field at the endpoint.
    Used in Eq. 4 to turn the negative definite Jacobian network into a vector field with that Jacobian. Integration by parts along the path yields exact equality, but the paper does not show the computation.
  • standard math Contraction is invariant under diffeomorphic coordinate changes.
    Stated as Theorem 1 (Manchester and Slotine 2017) and used to transfer latent-space contraction to the data space through the injective decoder.
  • domain assumption The zero-padding injective flow decoder is a diffeomorphism onto its image.
    Taken from Brehmer and Cranmer (2020); the paper assumes the decoder mu is injective so that contraction transfers from latent to ambient space (Sec. 3.2).
  • domain assumption The modulation matrix G(x) preserves contraction of the modulated vector field.
    The paper states this follows from Huber et al. (2019, 2022) but does not prove it for the learned Riemannian pullback metric setting of Sec. 3.4.
  • ad hoc to paper Out-of-data-support regions are unsafe and should be avoided.
    A modeling choice used to define Riemannian safety regions; it is reasonable for many robotics settings but is not derived from data or first principles.
invented entities (1)
  • Riemannian safety region in latent space
    purpose: Designates obstacles and out-of-data-support areas as unsafe so the modulated vector field avoids them.
    Constructed from the volume of the pullback Riemannian metric; it is a modeling construct with no falsifiable prediction outside the paper.

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Pith. "Pith review of Extended Neural Contractive Dynamical Systems: On Multiple Tasks and Riemannian Safety Regions." pith.science (2026). https://pith.science/paper/QYSMHLQ6

@misc{pith2026241111405,
  author       = {Pith},
  title        = {Pith review of: Extended Neural Contractive Dynamical Systems: On Multiple Tasks and Riemannian Safety Regions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QYSMHLQ6}},
  note         = {Machine review of arXiv:2411.11405}
}
read the original abstract

Stability guarantees are crucial when ensuring that a fully autonomous robot does not take undesirable or potentially harmful actions. We recently proposed the Neural Contractive Dynamical Systems (NCDS), which is a neural network architecture that guarantees contractive stability. With this, learning-from-demonstrations approaches can trivially provide stability guarantees. However, our early work left several unanswered questions, which we here address. Beyond providing an in-depth explanation of NCDS, this paper extends the framework with more careful regularization, a conditional variant of the framework for handling multiple tasks, and an uncertainty-driven approach to latent obstacle avoidance. Experiments verify that the developed system has the flexibility of ordinary neural networks while providing the stability guarantees needed for autonomous robotics.

Figures

Figures reproduced from arXiv: 2411.11405 by the authors.

Figure 1
Figure 1. Robot motion generated for a grasping skill using a Conditional Neural Contractive Dynamical System (CNCDS). The robot motion is conditioned on an input image of the object placed on a desk. CNCDS also learns a Riemannian manifold from demonstrations, representing a safety region to navigate through. dynamical systems (Tsukamoto and Chung 2021a; Dawson et al. 2023; Jaffe et al. 2024), was designed to learn a single … view at source ↗
Figure 2
Figure 2. The effect of eigenvalue differences and asymmetry of the Jacobian on the contraction behavior of the vector field. Left: In a contractive dynamical system with a symmetric Jacobian, when the eigenvalues are equal, the system contracts uniformly in all directions. Middle: In a contractive dynamical system with a symmetric Jacobian, when the eigenvalues are different, the system contracts more rapidly in the directio… view at source ↗
Figure 4
Figure 4. Comparison of 2D vector fields learned by using the following regularization approaches: (a) constant regularization, (b) state-independent regularization (using basic optimization), (c) state-dependent regularization (using a neural network), and (d) eigenvalue regularization. In the top row, the gray background illustrates the learned vector fields, with colored trajectories (integral curves) starting from the ini… view at source ↗
Figures from the paper (27 more)
Figure 5
Figure 5. Figure 5: Comparison of 2D vector fields learned with different Jacobian formulations. Left: Vector field obtained using the symmetric Jacobian formulation. Middle: Vector field obtained using the asymmetric Jacobian formulation. The learned vector field (grey) and demonstration…
Figure 6
Figure 6. Figure 6: Comparison between asymmetric and symmetric Jacobians in learning contractive dynamical systems. Left: Generalization behavior of a system with a symmetric Jacobian, visualized using integral curves originating from a 10 × 10 equidistant grid. The yellow region represe…
Figure 7
Figure 7. Figure 7: The distribution of the duration, in time steps, that each integral curve remained within the demonstration region across four different regularization methods: Constant, State Independent, State Dependent, and Eigenvalue Regularization. A higher count of time steps in…
Figure 8
Figure 8. Figure 8: CNCDS on 2D trajectories from the LASA handwriting dataset. Plots (a) and (b) show CNCDS conditioned on the trajectory shape. Plots (c) and (d) display conditioning on the trajectory target, where the conditional value is chosen to be 0 and 1, respectively. This is sho…
Figure 9
Figure 9. Figure 9: The obstacle locally reshapes the learned vector field using the modulation matrix. The gray contours represent the learned vector field, black trajectories depict the demonstrations, and orange/green trajectories are integral curves starting from both initial points o…
Figure 10
Figure 10. Figure 10: Architecture overview: a single iteration of NCDS simultaneously generating position and orientation dynamics. (A) VAE (pink box): The encoder processes the concatenated position-orientation data pt, yielding a resulting vector that is subsequently divided into two co…
Figure 11
Figure 11. Figure 11: Left: Aspects of the Lie group SO(3). Right: An illustration of the function b(x) in two dimensions. 3.3.1 Orientation parameterization. Three-dimensional spatial orientations can be represented in several ways, including Euler angles, unit quaternions, and rotation m…
Figure 12
Figure 12. Figure 12: Modulated contractive dynamical systems using a pullback metric derived from the decoder’s Jacobian. corresponds to the set of pure imaginary quaternions, which can be represented as 3D vectors. A quaternion q ∈ S3 is typically expressed as q = q0 + qxi + qyj + qzk, w…
Figure 13
Figure 13. Figure 13: Illustration of the computation of the obstacle avoidance distance field for designing Riemannian safety regions. From left: (1) Obstacle representation (circle), (2) Metric volume around obstacle, (3) Metric inversion to compute distance map and problem with initial …
Figure 14
Figure 14. Figure 14: Visualization of the LASA-2D dataset: Gray contours represent the learned vector field, black trajectories depict demonstrations, and orange/green trajectories illustrate integral curves starting from the initial points of the demonstrations and plot corners. The mage…
Figure 15
Figure 15. Figure 15: Robot experiments. The left two panels show robot experiments with orange and red paths illustrating unperturbed and perturbed motion. The first panel shows a learned vector field in R 3 with a constant orientation. In the second panel, the vector field expands to R 3…
Figure 16
Figure 16. Figure 16: Average distance between random nearby trajectories over time for LASA-2D. Only NCDS monotonically decreases, i.e. it is the only contractive method. reaffirms the inherent computational intensity tied to Jacobian￾based computations. This table also shows that our VAE…
Figure 17
Figure 17. Figure 17: integral curves generated using different methods on LASA-8D. Black curves are training data, while yellow are the learned integral curves. To construct the 8D dataset, we have concatenated 4 different 2D datasets [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]
Figure 18
Figure 18. Figure 18: Comparison of vector fields generated by NCDS and ELCD (Jaffe et al. 2024). The left, middle, and right columns depict the integral curves produced by ELCD, vanilla NCDS, and NCDS with state-independent regularization, respectively. A total of 100 integral curves, eac…
Figure 19
Figure 19. Figure 19: Comparison of contractivity across methods. The top and middle plots show the number of trajectory points inside the demonstration region for models trained on Angle and Sine datasets, respectively, where higher values indicate stronger contraction. The bottom table p…
Figure 20
Figure 20. Figure 20: Generalization under deviation from the decoder’s manifold using the 8D LASA dataset. Red circles indicate initial positions, the orange dashed line shows the transition vector to the manifold (green circle), and the blue curve represents the integral curve. while kee…
Figure 21
Figure 21. Figure 21: Obstacle avoidance via a Riemannian pullback metric. Left: The robot performs a grasping task without obstacle avoidance. Middle: The robot navigates around the obstacle by avoiding both the obstacle and the unsafe regions of the manifold. Right: The latent modulated …
Figure 22
Figure 22. Figure 22: Left: Latent integral curves with NCDS trained in joint space R 44 . Right: Latent integral curves with NCDS trained on full human motion space: R 44 × R 3 × SO(3). The background depicts the contours of the latent vector field, green and yellow curves depict the inte…
Figure 23
Figure 23. Figure 23: Grasping and dropping actions learned by the CNCDS for a soft object in three positions, conditioned on image input. The CNCDS is trained using cross-entropy loss, velocity reconstruction, and regularization with ResNet18 image embeddings [PITH_FULL_IMAGE:figures/ful…
Figure 24
Figure 24. Figure 24: Left: Generated Human Motion with NCDS trained in joint space R 44: The upper row depicts the progressive evolution of the demonstration motion from left to right. Meanwhile, the middle and bottom rows illustrate the motions generated by NCDS when the initial point is…
Figure 25
Figure 25. Figure 25: Integral curves generated by NCDS trained using different regularization term ϵ. (a) Contractive (b) Unconstrained (c) MLP (d) Neural ODE [PITH_FULL_IMAGE:figures/full_fig_p023_25.png]
Figure 26
Figure 26. Figure 26: Integral curves generated under the Neural Contractive Dynamical Systems (NCDS) setting, along with baseline comparisons using Multilayer Perceptron (MLP) and Neural Ordinary Differential Equation (NeuralODE) models. (a) Tanh (b) Softplus (c) Sigmoid [PITH_FULL_IMAGE…
Figure 27
Figure 27. Figure 27: Integral curves generated by NCDS trained using different activation functions. highlight the effect of the contraction constraint on the behavior of the dynamical system. The MLP baseline uses a neural network with 2 hidden layers each with 100 neurons with Tanh acti…
Figure 28
Figure 28. Figure 28: Learned vector field with zero contraction spread. Left: The contraction spread, scaled by 1000, highlights minimal eigenvalue differences. Right: Orange integral curves closely follow the training demonstrations (black), with gray contours outlining the learned vecto…
Figure 29
Figure 29. Figure 29: The behavior of the elements of the matrix D in response to distance from obstacle. 6.2 Modulation scaling factor To compute the weight α, we discretize the data manifold with an equidistant mesh grid in the latent space Z, followed by the computation of the correspon…
Figure 30
Figure 30. Figure 30: Comparison of 2D vector fields learned by using the following regularization approaches: (a) constant regularization, (b) state-independent regularization (using basic optimization), (c) state-dependent regularization (using a neural network), and (d) eigenvalue regul…
Figure 31
Figure 31. Figure 31: Eigenvalue metric heatmaps of the learned contractive dynamics shown in [PITH_FULL_IMAGE:figures/full_fig_p029_31.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.