Elastic ODYN: Differentiable Optimization for Infeasible Control and Learning in Robotics
Aristotelis Papatheodorou, Jose Rojas, Ioannis Havoutis, Carlos Mastalli
Abstract
Robotic systems routinely encounter conflicting objectives, modeling errors, and degenerate contact conditions that render quadratic programs (QPs) infeasible. Yet most optimization solvers and differentiable QP layers assume feasibility, leading to numerical failures, unstable gradients, or solver breakdown when constraints cannot be simultaneously satisfied. We present Elastic ODYN, a primal--dual non-interior-point QP solver that handles infeasibility through smooth squared- elastic relaxations. The resulting formulation remains well posed under ill-conditioning and degeneracy, supports warm starting, and converges to closest-to-feasible solutions when no feasible point exists. A lightweight refinement stage recovers physically meaningful dual variables from the elastic solution. Building on this framework, we develop Elastic OdynLayer, a differentiable QP layer with stable gradients under infeasibility, and Elastic OdynSQP, an infeasibility-aware SQP method that resolves inconsistent subproblems and intrinsically infeasible optimal control tasks through selective constraint relaxation. We evaluate the framework on benchmark QPs, singular contact mechanics, differentiable parameter identification, and quadrupedal and humanoid trajectory optimization. Across all settings, Elastic ODYN consistently outperforms state-of-the-art elastic QP solvers in robustness, warm-start performance, and convergence reliability, enabling optimization, simulation, control, and learning beyond the feasibility assumptions of existing methods.
AI Impact Assessments
(1 models)Scientific Impact Assessment: Elastic ODYN
1. Core Contribution
Elastic ODYN addresses a genuine and pervasive problem in optimization-based robotics: what happens when quadratic programs (QPs) become infeasible due to conflicting constraints, modeling errors, or degenerate contact conditions. The paper contributes a unified framework with three interconnected components:
1. Elastic ODYN: A primal-dual non-interior-point QP solver using smooth squared-ℓ₂ elastic relaxations that remains well-posed under infeasibility, ill-conditioning, and degeneracy.
2. Elastic OdynLayer: A differentiable QP layer that provides stable gradients even when no feasible point exists.
3. Elastic OdynSQP: An infeasibility-aware SQP method with selective constraint relaxation for nonlinear optimal control.
The key insight is combining Moreau-Yosida regularization (squared-ℓ₂ penalties) with a non-interior-point method framework, yielding a C¹-smooth formulation that naturally supports both warm-starting and implicit differentiation—properties that ℓ₁-based elastic methods or interior-point methods cannot simultaneously provide.
2. Methodological Rigor
The mathematical formulation is carefully developed. The penalty-barrier formulation (Eq. 12-14) elegantly combines elastic relaxation variables with proximal primal-dual regularization, and the closed-form elimination of consensus and elastic variables (Section IV-C) is a clean algebraic reduction that maintains the hard-constrained problem dimensions. The stopping criteria (Section IV-D) are thoughtfully designed to certify convergence to closest-feasible solutions via least-violation optimality conditions.
Strengths in rigor:
Concerns:
3. Potential Impact
The framework addresses a real gap at the intersection of optimization, robotics, and machine learning:
The potential impact is substantial for the robotics optimization community, though the paper's reach into broader ML/AI may be limited by the specialized problem setting.
4. Timeliness & Relevance
This work is highly timely. The convergence of differentiable simulation, learning-based control, and contact-rich manipulation has created an acute need for optimization solvers that gracefully handle infeasibility. Current tools (OptNet, CvxpyLayers) assume feasibility; QpLayer handles infeasibility but through non-smooth ECJ framework. The smooth differentiable approach fills a clear niche.
The selective outer relaxation strategy for SQP (choosing which constraint categories to relax) is practically valuable for trajectory optimization, where users must balance physical fidelity against computational tractability.
5. Strengths & Limitations
Key Strengths:
Notable Limitations:
Additional Observations:
The paper promises open-source release upon acceptance, which would significantly enhance impact. The merit function design for SQP (Section V-E) combining ℓ₂ elastic and ℓ₁ hard penalties is an interesting hybrid but its theoretical properties (e.g., exactness conditions) are not analyzed.
Overall, this is a well-conceived systems paper that addresses a genuine need with a mathematically coherent framework. The experimental validation, while not exhaustive, covers diverse relevant scenarios. The main weaknesses are in theoretical depth and the scope of empirical comparisons, particularly for the differentiable optimization component.
Generated Jun 16, 2026
Comparison History (37)
Paper 2 likely has higher impact: it targets autonomous driving safety, a highly timely and societally critical domain, and proposes a scalable post-training paradigm using synthesized long-tail interactive scenarios, with demonstrated gains on public benchmarks and production-scale deployment plus code release—factors that typically drive broad adoption and cross-field influence (generative modeling, RL, safety, simulation). Paper 1 is methodologically strong and valuable for robotics optimization/differentiable control, but its core contribution is more specialized and incremental relative to existing elastic/QP-layer lines, with narrower immediate real-world reach.
Paper 1 explores a timely and high-impact question: leveraging frontier video generation models (Veo-3) for generalizable robot manipulation. It proposes a novel hierarchical framework combining video models with VLAs, addressing a broadly relevant problem in embodied AI. The approach of using video models as motion planners with inverse dynamics is innovative and has significant implications as video models improve. Paper 2 makes solid contributions to differentiable optimization under infeasibility, but addresses a more specialized technical problem. Paper 1's broader relevance across robotics, foundation models, and generalization gives it higher potential impact.
Elastic ODYN addresses a fundamental and pervasive problem in robotics optimization—handling infeasibility in QP solvers—with rigorous mathematical foundations, differentiable layers, and broad applicability across control, simulation, learning, and trajectory optimization for various robot morphologies. Its contributions are foundational and applicable across many subfields. Paper 2, while novel in combining LLMs with conformal prediction for multi-agent coordination, addresses a narrower problem (MM-EQA) with less generalizable methodology and relies heavily on current LLM capabilities, limiting its long-term foundational impact.
Paper 2 ($μ_0$) is likely to have higher scientific impact due to its broader, more scalable framing: an embodiment-agnostic 3D trace representation that can transfer across robots and leverage diverse Internet video, with a pipeline (TraceExtract) for automatic 3D supervision and language alignment. This targets a timely, high-visibility direction (foundation/world models for robotics) with wide applicability across manipulation, representation learning, and vision-language-robotics. Paper 1 is methodologically strong and valuable for robust control/optimization, but its impact is narrower to QP/SQP-based robotics pipelines.
Paper 2 has broader cross-field impact and timeliness: a robust, differentiable QP solver/layer handling infeasibility addresses a pervasive failure mode in optimization-based robotics, control, simulation, and learning, and can be adopted widely beyond specific hardware. Its methodological contribution (primal–dual solver, stable gradients, SQP integration, dual refinement) is general and likely to become infrastructure for many systems. Paper 1 is highly innovative and valuable for tactile manipulation, but its impact is more domain- and data/ecosystem-dependent, with narrower immediate applicability than a foundational optimization tool.
Paper 2 addresses a fundamental and pervasive mathematical problem in robotics and machine learning: infeasibility and instability in Quadratic Programs (QPs) and differentiable optimization. While Paper 1 makes strong contributions to visuomotor policies and imitation learning, Paper 2's algorithmic advancements can be integrated into a much broader array of domains, including control, simulation, and trajectory optimization for complex systems like humanoids. This foundational improvement to widely used optimization layers gives it a higher potential for broad, cross-disciplinary scientific impact.
Paper 1 likely has higher scientific impact: it addresses a pervasive failure mode (QP infeasibility/degeneracy) across robotics optimization, control, simulation, and differentiable learning, offering a solver + differentiable layer + SQP framework with broad applicability. Its contribution is methodologically substantial (well-posed elastic formulation, dual recovery, warm-starting, stable gradients) and timely given the widespread use of differentiable optimization layers and contact-rich control. Paper 2 is novel and rigorous in regret-guided replanning, but is narrower in scope and appears to rely on stronger modeling assumptions (bounded drift) with more limited cross-domain adoption.
Paper 1 likely has higher scientific impact due to broader methodological novelty and cross-domain applicability: it introduces a robust, differentiable QP/SQP framework that remains stable under infeasibility—an issue pervasive across robotics, optimal control, simulation, and learning. If rigorous and well-validated, such a solver/layer can become infrastructure adopted by many downstream systems and research areas. Paper 2 is timely and compelling with strong real-world demonstrations, but its contributions are more application-specific (quadrotor fall recovery) and may generalize less broadly than a foundational optimization method.
Elastic ODYN addresses a fundamental and broadly impactful problem in robotics optimization—handling infeasibility in QPs—which affects control, simulation, learning, and trajectory optimization across many robotic platforms. It offers rigorous mathematical contributions (differentiable QP layers with stable gradients under infeasibility, infeasibility-aware SQP), demonstrates broad applicability (contact mechanics, parameter identification, humanoid/quadruped locomotion), and advances the theoretical foundations of optimization in robotics. Paper 2 presents a useful engineering toolchain for sim-to-real data pipelines but is more application-specific, less methodologically novel, and tied to a particular commercial platform (JD Cloud), limiting its broader scientific influence.
Paper 1 offers a fundamental algorithmic contribution by addressing the ubiquitous issue of infeasible QPs in robotics through a novel differentiable solver. Its theoretical rigor and broad applicability across simulation, control, trajectory optimization, and learning provide a foundational tool for the field. In contrast, while Paper 2 tackles a highly relevant practical issue in imitation learning execution, it acts more as an applied, plug-and-play fix rather than a foundational methodological advancement, making Paper 1's potential scientific impact broader and deeper.