Elastic ODYN: Differentiable Optimization for Infeasible Control and Learning in Robotics

arXiv:2606.16564 · cs.RO, cs.LG · Submitted 2026-06-15 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: I'm Rosa, and with me are Dev and Taro, guest researcher.

Dev: Today's paper: "Elastic ODYN: Differentiable Optimization for Infeasible Control and Learning in Robotics".

Rosa: We present ELASTIC ODYN, a primal–dual non-interior-point QP solver that handles infeasibility through smooth squaredl2 elastic relaxations.

Dev: First, who's behind it and why it matters.

Title and authors: Rosa: So we're looking at the paper titled "Elastic ODYN: Differentiable Optimization for Infeasible Control and Learning in Robotics," which seems to address a really fundamental problem in optimization where things just don't work out as expected.

Dev: Exactly, Rosa, it tackles how robotic systems frequently run into conflicting objectives or modeling errors that make quadratic programs completely infeasible, which is a huge issue because most standard solvers and differentiable layers just crash or produce unstable gradients when they hit those impossible constraints.

Taro: I'm interested in how this framework handles those moments when the real world misbehaves, like when a robot tries to do something that's physically impossible given its current state.

Rosa: Well, the paper introduces ELASTIC ODYN as a primal-dual non-interior-point QP solver that manages these infeasibilities using smooth squared two elastic relaxations, which keeps the formulation well-posed even when things are ill-conditioned or degenerate.

Dev: That sounds promising for real-time control loops because it supports warm starting and can converge to a closest feasible solution even when no feasible point actually exists, which is something traditional methods struggle with.

Taro: If we look at the core idea, they use a computable constraint violation term, viol(x) = Ax - b, x - I(x), to serve as a tractable surrogate for the distance to the feasible set F, even though that true distance calculation is usually intractable.

Rosa: That surrogate approach is smart because it allows them to work around the intractability of calculating the true distance, which helps them manage those tough scenarios in robotics.

Dev: I'm also paying attention to how they handle the mathematical structure when things are infeasible; they show that when primal infeasibility occurs, the KKT inclusion fails because the normal cone becomes empty, meaning there are no compatible dual multipliers.

Taro: That failure point is critical for autonomy research because it shows us exactly where our current optimization methods break down when we encounter unexpected environmental conflicts or kinematic limits.

Rosa: Beyond just solving the QP, the paper develops ELASTIC ODYNLAYER, which is a differentiable QP layer that maintains stable gradients even under infeasibility, and ELASTIC ODYNSQP, an SQP method that specifically resolves inconsistent subproblems through selective constraint relaxation.

Dev: The idea of a differentiable layer that doesn't break when things go wrong during training is huge for learning-based methods; it means we can train policies even if the intermediate steps aren't strictly feasible, which is a big win for complex dynamics.

Taro: That differentiability under infeasibility opens up avenues for learning in nonsmooth contact dynamics, like identifying restitution coefficients where standard feasibility-dependent layers are usually impossible to use effectively.

Rosa: And then they include a lightweight refinement stage that recovers physically meaningful dual variables from the elastic solution, which is important because those multipliers tell us something about the forces involved in the problem.

Dev: Recovering those interpretable dual variables is key for engineers because it lets us actually understand what those multipliers represent in terms of physical constraints or forces during operation.

Taro: If we can get physically meaningful multipliers, it helps ground the optimization results and makes the resulting control actions more trustworthy when deployed in complex autonomous systems.

Rosa: So, to wrap up this discussion on "Elastic ODYN: Differentiable Optimization for Infeasible Control and Learning in Robotics," we see a framework that robustly handles infeasibility using elastic relaxations, coupled with tools like ELASTIC ODYNLAYER and ELASTIC ODYNSQP.

Dev: The main implication for me is the stability; it offers a way to keep the optimization loop running reliably even when the constraints are contradictory, which means less time spent debugging solver failures in control pipelines.

Taro: From an autonomy standpoint, I think this gives us a much better toolset for planning in environments where we have conflicting requirements or singular contact mechanics that usually lead to solver breakdown.

Rosa: It really shows how we can build systems that are more resilient when dealing with the messy realities of physical interaction and modeling imperfections in robotics.

Dev: I think the warm-starting capability mentioned is a practical necessity for me, especially in continuous control, because it means we don't have to restart from scratch every single time the prediction horizon shifts slightly.

Taro: We need to see how this translates when we move beyond simple QPs into more complex, mission-oriented tasks where the constraints are constantly changing based on perception.

Rosa: That brings us to our final thoughts on this paper; it provides a unified and computationally efficient way to deal with infeasibility that works across multiple optimization tasks in robotics.

Dev: It’s a solid piece of engineering because it tackles the numerical stability issues head-on without forcing every single problem into an interior-point framework that can be too slow or fragile for real-time use.

Taro: I think the ability to recover those physically meaningful dual variables really elevates this work beyond just a clever numerical trick; it gives us insight into the physics of why things are constrained in the first place.

Rosa: Overall, "Elastic ODYN: Differentiable Optimization for Infeasible Control and Learning in Robotics" offers a robust framework for when optimization hits walls, providing stability and interpretability where standard methods often fail.

The paper's summary: Rosa: So we're looking at the summary of "Elastic ODYN: Differentiable Optimization for Infeasible Control and Learning in Robotics," which essentially boils down to introducing a robust framework for tackling quadratic programming problems that simply don't have feasible solutions, often caused by modeling errors or conflicting requirements.

Dev: Right, Rosa, and the core idea is using these smooth two elastic relaxations to manage those infeasibilities without breaking the optimization loop, which is vital for keeping things running smoothly in control systems.

Taro: And what I find particularly interesting from the summary is how they tackle that situation where a robot's planned movement hits a physical wall or constraint that makes the math impossible to satisfy directly, moving beyond just stopping and trying again.

Rosa: Exactly, Taro, because they introduce ELASTIC ODYN as this primal-dual non-interior-point solver that keeps things well-posed even when the problem is degenerate or ill-conditioned.

Dev: That stability is what catches my attention; if we can have a solver that supports warm starting and still finds a close solution even when it's infeasible, that drastically reduces the latency we worry about in real-time control.

Taro: And the paper also details their extensions, like ELASTIC ODYNLAYER for differentiable layers and ELASTIC ODYNSQP for inconsistent subproblems, which means we can build learning systems that are more forgiving of the messy physical world.

Rosa: That's where I get really excited; imagine training a policy or a dynamics model where the underlying physics might be slightly off or encounter weird contact scenarios, and you don't have to halt the entire training process because of one impossible constraint violation.

Dev: From an engineering standpoint, being able to propagate gradients through these layers even when no feasible solution exists means we can train models for things like nonsmooth contact dynamics where standard feasibility-dependent QP layers just fail completely.

Taro: That opens up possibilities for learning in areas like identifying restitution coefficients from impact data, which is a problem that has always been tricky because it requires knowing the exact state of collision.

Rosa: And the refinement stage that recovers physically interpretable dual variables is a big deal because it doesn't just give us an answer; it gives us something meaningful about the forces or constraints involved in reaching that closest-to-feasible point.

Dev: I agree, those multipliers are crucial for interpreting how much pressure or force is actually being applied at a contact point, which is exactly what we need when designing physical controllers.

Taro: So, it seems like the real impact here is enabling more resilient autonomous systems that can handle unpredictable physical interactions without crashing or giving nonsensical control outputs when things don't go according to the perfect mathematical model.

Rosa: It really does show how this framework helps bridge the gap between abstract optimization theory and the messy, constrained reality of robotics and learning.

Dev: So we've covered how this handles infeasibility and provides differentiability under those tough conditions; what I want to know next is if these methods are practical for long-term deployment on actual robotic hardware, Rosa?

The paper's improvements: Taro: So we've gone over how ELASTIC ODYN manages the infeasibility in QPs, and now we're looking at what they suggest to make it even better, focusing on these specific improvements like ELASTIC ODYNLAYER and ELASTIC ODYNSQP.

Rosa: Right, Taro; the paper proposes a whole suite of tools: not just solving the problem robustly, but also providing a differentiable layer that works under infeasibility for learning tasks, which is huge.

Dev: I’m looking at ELASTIC ODYNLAYER specifically because stable gradients during infeasibility are exactly what we need when training policies or dynamics models where the constraints might be violated during intermediate steps.

Taro: And then there's ELASTIC ODYNSQP, which is an SQP method designed to specifically handle inconsistent subproblems by using selective constraint relaxation, which addresses those tricky cases where the problem itself is fundamentally contradictory.

Rosa: That selective relaxation sounds like a clever way to resolve the inconsistency directly within the optimization framework instead of just trying to penalize it away.

Dev: It’s important that this whole system supports warm starting efficiently, because if we're running continuous control loops at high rates, we need to be able to quickly re-solve the problem without losing precious loop time.

Taro: I think these enhancements suggest a future where AI systems can train themselves more effectively in physical simulations, even when those simulations are based on imperfect or nonsmooth contact models that lead to infeasibility.

Rosa: And the dual recovery refinement stage, which we talked about before, is also an improvement because it gives us actual numbers for the Lagrange multipliers so we can actually understand what those constraints mean in terms of physical forces.

Dev: That interpretation is key; if we're designing a whole-body humanoid controller, knowing what those dual variables represent helps us tune the control gains to be physically realistic and safe.

Taro: So, the implication here is that we can push AI into more complex physical interaction domains where the physics are messy, not just neat quadratic problems with perfect constraints.

Rosa: It really suggests that this approach isn't just a numerical trick for solving QPs; it’s a foundational shift toward building AI models that are robust to the inherent uncertainties of real-world physical interaction.

Dev: I wonder about the deployment timeline; while it solves the mathematical problem, we need to see if these solvers can maintain their performance when running on embedded hardware at a very high frequency, say one hundred Hertz or more.

Taro: That’s a fair point, Dev; the complexity of these elastic relaxations means there’s always a trade-off between accuracy and computational speed that we need to keep in mind for real-time autonomy.

Rosa: So, while the theoretical framework is incredibly robust and powerful for handling infeasibility, we're still waiting to see how smoothly it integrates into the existing high-speed hardware pipelines of field robotics.

Conclusion: Rosa: So we're wrapping up our discussion on "Elastic ODYN: Differentiable Optimization for Infeasible Control and Learning in Robotics," which really shows how this new framework tackles infeasibility using smooth squared two elastic relaxations to keep optimization stable.

Dev: I agree, Rosa, it’s a solid piece of engineering that offers a way to handle those tricky constraint violations without the whole control loop collapsing under numerical stress.

Taro: From my viewpoint as an autonomy researcher, the real impact is how this gives us tools to plan in environments where the physical constraints are constantly shifting or when we encounter unexpected contact mechanics during navigation.

Rosa: Exactly, Taro; it moves us closer to building AI systems that can operate reliably in messy physical interactions without needing perfect mathematical modeling upfront.

Dev: I’m still thinking about the practical side; how long can we expect this solver to run reliably on actual embedded hardware in a demanding, real-time control scenario?

Taro: That’s a valid concern, Dev; the computational overhead of these elastic mechanisms is something we need to watch closely when moving from simulation to live robotics.

Rosa: We certainly do have those questions about deployment feasibility, but this work sets up a really strong foundation for more advanced tasks in robotics and learning.

Dev: I'm ready for the next paper discussion; I want to see how these concepts translate into tangible latency reductions in my control loops.

University of Oxford · Robot Motor Intelligence (RoMI) Lab

cs.RO, cs.LG

Submitted: 2026-06-15

Updated: 2026-09-29

Comments: 8 pages, 5 figures, 3 tables

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 89/100

The gist: We present ELASTIC ODYN, a primal–dual non-interior-point QP solver that handles infeasibility through smooth squaredl2 elastic relaxations.

Key concepts

Elastic ODYN
A primal-dual non-interior-point QP solver that manages optimization problems with infeasibility using smooth squared two elastic relaxations to keep the formulation well-posed, even when the problem is ill-conditioned or degenerate.
Constraint Violation Term
A computable surrogate used to estimate the distance to a feasible set F, defined as viol(x) = |Ax - b| or |x - I(x)|. This allows the solver to work around the intractability of calculating the true distance to feasibility.
ELASTIC ODYNLAYER
A differentiable QP layer developed by the paper that maintains stable gradients even when infeasibility occurs. This is crucial for training learning-based methods where intermediate steps might not strictly satisfy constraints.
Physically Meaningful Dual Variables
Dual variables recovered from the elastic solution that provide insight into the forces or constraints involved in reaching a closest-to-feasible point. These multipliers help engineers understand physical forces during operation.

Terminology

Summary

We present ELASTIC ODYN, a primal–dual non-interior-point QP solver that handles infeasibility through smooth squaredl2 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.

This paper introduces ELASTIC ODYN∗, a framework for robust and differentiable quadratic programming under infeasibility. Our main contributions are: (i) Elastic ODYN. We develop ELASTIC ODYN, a smooth l2-elastic non-interior point QP solver based on ODYN [6], for robustly handling infeasible, degenerate, and ill-conditioned QPs. (ii) Dual recovery. We introduce a refinement procedure that recovers interpretable dual variables for infeasible QPs. (iii) Differentiable optimization under infeasibility. We develop ELASTIC ODYNLAYER, a differentiable QP layer with stable gradients for both feasible and infeasible problems. (iv) Infeasibility-aware SQP. We develop ELASTIC ODYNSQP, an SQP framework that resolves inconsistent subproblems and intrinsically infeasible optimal control tasks through selective constraint relaxation.

Quadratic programming (QP) is a fundamental tool across science and engineering. In robotics and artificial intelligence (AI) optimization problems are often large-scale and numerically stiff, placing stringent requirements on solver robustness, scalability, and numerical stability. In practice, feasibility is frequently violated to modeling errors, external disturbances, or conflicting task requirements. As a result of this violation in control pipelines may themselves become infeasible. Model predictive control (MPC) computes control actions in a principled way by repeatedly solving structured quadratic programs. However, MPC scales poorly and becomes increasingly ill-conditioned with longer prediction horizons. Moreover, MPC solvers are not designed to handle infeasible constraint sets gracefully, which can lead to failure or unpredictable behavior in practice. Within sequential quadratic programming (SQP) methods, infeasibility is typically addressed through restoration stages. In contrast, elastic mechanisms provide a unified and computationally efficient means of recovering feasibility while resolving the problem.

Differentiable optimization has emerged as a promising paradigm for learning-based methods, where the forward pass solves a constrained problem and the backward pass propagates loss gradients through its solution. These gradients are governed by the Karush–Kuhn–Tucker (KKT) system, which couples primal and dual variables and enforces stationarity and complementarity, but not primal feasibility, since intermediate gradient-based iterates may be infeasible. As a result, differentiable optimization layers are highly sensitive to infeasibility and ill-conditioning.

We consider convex quadratic programs of the form min x T Q x + c T x subject to A x = b, G x ≤ h, with primal variables x ∈ R n, symmetric positive semidefinite cost matrix Q ∈ R n×n, linear cost c ∈ R n, and constraints defined by A ∈ R m×n, b ∈ R m for the equalities and G ∈ R p×n, h ∈ R p for the inequalities. The KKT conditions admit the equivalent variational form 0 = Qx + c + NF (x), with feasible set F:= F:= F =

(x1, x2) ∈ R2

x1 = 1, x1 = −2, x2 = 1, x2 = 103

G x ≤ h

A point is primal infeasible if it violates at least one constraint. The computable constraint violation is defined as viol(x) = max∥Ax − b∥, ∥x − ΠI(x)∥. For x ∈ F/, the variational inclusion cannot be satisfied, meaning the KKT inclusion cannot hold: 0 ∈/ Qx + c + NF (x), because the normal cone is empty, i.e, NF (x) = ∅ for x ∈/ F.

Dual feasibility concerns the existence of Lagrange multipliers (y, z) compatible with the normal-cone representation of Eq. (3).

Improvements for AI systems

Here are the specific improvements to AI systems achievable by implementing ELASTIC ODYN and its derivatives, based on the provided paper:

) 1. Robust Planning for High-Dimensional Robotic Systems (Via ELASTIC ODYNSQP):

The system can reliably solve complex, high-dimensional optimal control problems for legged robots (quadrupeds/humanoids) even when the constraints are inherently conflicting or lead to inconsistent subproblems.

  • Specific Capability: Enables real-time motion planning and trajectory optimization under challenging conditions like singular contact mechanics or contradictory task requirements (e.g., simultaneously maintaining a specific posture while navigating an obstacle).

  • Benefit: Prevents solver failure, which is common in Model Predictive Control (MPC) or Sequential Quadratic Programming (SQP) when constraints are violated, leading to stable control and predictable behavior during dynamic maneuvers.

) 2. Differentiable Learning for Infeasible Dynamics and Parameter Identification (Via ELASTIC ODYNLAYER):

The system can be integrated into reinforcement learning or differentiable physics-based simulators where the underlying dynamics or constraints are not perfectly known or may lead to infeasibility during training.

  • Specific Capability: Allows the propagation of loss gradients through optimization layers even when no feasible solution exists, by treating constraint violations (via elastic relaxation variables) as a differentiable signal. This enables learning in nonsmooth contact dynamics (e.g., restitution coefficient identification).

  • Benefit: Enables the training of neural network policies or dynamics models for complex physical interactions (like impact/restitution) that are traditionally difficult or impossible to train using standard, feasibility-dependent QP layers.

) 3. Robust Contact Simulation and Impulse Modeling (Via ELASTIC ODYN):

The system can accurately model contact mechanics at the level of acceleration and impulse, resolving geometric degeneracies that cause traditional solvers to fail.

  • Specific Capability: Resolves singularities in contact geometry (e.g., rank-deficient Jacobians during sustained contact or impact) by admitting small, physically meaningful penetrations and penalizing violations via smooth squared-l2 relaxations.

  • Benefit: Provides a stable simulation environment for rigid body dynamics and contact MPC, allowing robots to safely navigate narrow passages or undergo realistic impacts without the simulator crashing or producing non-physical forces.

) 4. Enhanced Warm-Start Performance in Real-Time Control (Via ELASTIC ODYN):

The system can efficiently handle dynamic changes in the optimization problem, such as those encountered during continuous MPC loops.

  • Specific Capability: Supports efficient warm starting from previous solutions, which is crucial for fast re-solving of control problems over time horizons.

  • Benefit: Minimizes computational latency in real-time applications, allowing the robot to react quickly to dynamic changes (e.g., unexpected disturbances) with minimal iteration counts per control cycle compared to methods requiring a full cold start.

) 5. Physically Interpretable Multipliers for Contact Forces (Via ELASTIC ODYN Refinement Stage):

The system can provide physically meaningful Lagrange multipliers, which are essential for interpreting the why behind the robot's actions in physics-based models.

  • Specific Capability: A lightweight refinement stage recovers dual variables from the elastic solution, yielding multipliers consistent with the closest feasible problem.

  • Benefit: Allows researchers and engineers to interpret contact forces or constraint sensitivities in a way that is physically meaningful (e.g., identifying actual contact forces rather than artifacts of penalty functions), which is critical for high-fidelity simulation and control design.

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- 2 elastic relaxations. The formulation remains well posed under ill-conditioning and degeneracy, supports warm starting, and converges to closest-to-feasible solutions, with lightweight refinement recovering physically meaningful dual variables. Building on this framework, we develop Elastic ODYNLayer, a differentiable QP layer with stable gradients under infeasibility, and Elastic OdynSQP, an SQP method that resolves inconsistent subproblems and intrinsically infeasible optimal control tasks through selective constraint elasticity. Across benchmark QPs, singular contact mechanics, differentiable parameter identification, and quadrupedal and humanoid trajectory optimization, Elastic ODYN outperforms state-of-the-art elastic QP solvers in robustness, warm-start performance, and convergence reliability, enabling optimization, simulation, control, and learning beyond standard feasibility assumptions.

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