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

This paper introduces Elastic ODYN, a differentiable primal-dual QP solver and framework that robustly handles infeasible control and learning tasks in robotics by employing smooth elastic relaxations to ensure numerical stability, convergence to closest-to-feasible solutions, and reliable gradient computation even under conflicting objectives and degenerate contact conditions.

Original authors: Aristotelis Papatheodorou, Jose Rojas, Ioannis Havoutis, Carlos Mastalli

Published 2026-06-16

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CC BY 4.0

Original authors: Aristotelis Papatheodorou, Jose Rojas, Ioannis Havoutis, Carlos Mastalli

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to plan a complex dance routine for a robot. You have a set of strict rules: "Don't fall over," "Keep your hand on the table," and "Move your foot exactly here." In the real world, robots often face situations where these rules contradict each other. Maybe the floor is slippery, or the robot's sensors are slightly off, or the rules themselves are impossible to satisfy all at once.

In the world of robotics math, this is called an infeasible problem. Most existing computer programs (called "solvers") are like rigid teachers: if a student breaks even one rule, the teacher shuts down, throws up their hands, and says, "I can't solve this!" This causes the robot to freeze, crash, or behave unpredictably.

The paper introduces a new tool called ELASTIC ODYN. Here is how it works, using simple analogies:

1. The "Rubber Band" Solution (Elastic Constraints)

Instead of treating rules as unbreakable steel walls, ELASTIC ODYN treats them like rubber bands.

Old Way: If a robot tries to touch a wall but can't quite reach it, the math breaks.
ELASTIC ODYN Way: If the robot can't perfectly touch the wall, the rubber band stretches just a tiny bit. The computer calculates the best possible solution where the rules are bent as little as possible, rather than giving up entirely. It finds the "closest possible" answer to the impossible request.

2. The "Smooth Operator" (Differentiability)

Robots today are learning from experience, similar to how a human learns to ride a bike by falling and adjusting. To learn, the computer needs to understand why it failed and how to fix it.

The Problem: When the rules are broken (infeasible), old math tools produce "jagged" or broken signals. It's like trying to drive a car with a steering wheel that suddenly snaps off when you hit a bump. The learning process crashes.
The Solution: ELASTIC ODYN is "smooth." Even when the robot breaks a rule, the math signal remains continuous and smooth. This allows the robot to learn from its mistakes even when the situation is messy or impossible, guiding it gently toward a better solution.

3. The "Refinement Stage" (Getting the Physics Right)

When the rubber bands stretch, the numbers the computer produces (called "dual variables") are a bit fuzzy. They tell you that a rule was broken, but not exactly how hard the robot pushed against the wall.

The Fix: The paper adds a quick "refinement stage." Think of this as a second, quick check. Once the computer finds the "stretched" solution, it runs a fast, lightweight calculation to translate those fuzzy numbers into real, physical forces (like how hard the robot's foot is pressing the ground). This ensures the robot understands the physics correctly, even if it had to stretch a rule to get there.

4. The "SQP" Upgrade (Handling Big, Complex Tasks)

The authors also built a bigger version called ELASTIC ODYN SQP. This is for planning long, complex sequences of movements (like a robot walking or dancing).

The Scenario: Sometimes, a robot's plan is impossible from the start (e.g., trying to walk through a wall).
The Magic: Instead of getting stuck, this tool selectively relaxes only the specific rules that are causing the conflict. It might say, "Okay, we can't hit that exact target, so let's aim for the spot right next to it," while keeping all other rules strict. This allows the robot to keep moving and finding a path, rather than freezing up.

Why This Matters

The paper tested this on difficult math problems, simulated robot contacts (like a disc hitting a tube), and real-world robot tasks (like a quadruped dog walking and a humanoid robot moving).

The Result:

Robustness: It doesn't crash when things go wrong.
Speed: It starts solving problems faster when given a "warm start" (a hint from a previous attempt).
Learning: It allows robots to learn and optimize even when their tasks are impossible or their models are slightly wrong.

In short, ELASTIC ODYN is a smarter, more flexible math engine that lets robots keep working and learning even when the world doesn't follow the perfect rules.

Technical Summary: ELASTIC ODYN

Problem Statement

Robotic systems frequently encounter scenarios where optimization problems become infeasible due to conflicting objectives, modeling errors, or degenerate contact conditions (e.g., rank-deficient contact Jacobians). Standard quadratic programming (QP) solvers and differentiable optimization layers typically assume feasibility. When constraints cannot be simultaneously satisfied, these methods often suffer from numerical failures, unstable gradients, or complete solver breakdown. This limitation hinders the robustness of Model Predictive Control (MPC), Sequential Quadratic Programming (SQP), and learning-based robotics pipelines, particularly in real-time, high-dimensional settings where intermediate iterates may be infeasible.

Methodology

The paper introduces ELASTIC ODYN, a framework designed to handle infeasibility, degeneracy, and ill-conditioning through smooth squared- $\ell_2$ elastic relaxations. The methodology is built upon three core components:

1. The ELASTIC ODYN Solver

This is a primal–dual non-interior-point QP solver. Unlike interior-point methods (IPMs) that require strict feasibility to follow a central path, ELASTIC ODYN operates outside the feasible set.

Elastic Formulation: Hard constraints ( $Ax=b, Gx \le h$ ) are relaxed using elastic variables ( $s_e, s_i$ ) with quadratic penalties in the objective function. This transforms the problem into an unconstrained form that remains well-posed even when the original constraints are contradictory.
Penalty–Barrier Approach: The solver utilizes a penalty–barrier formulation where a barrier function acts on a consensus variable rather than the slack variables directly. This allows iterates to leave the interior of the feasible region while maintaining a well-defined primal–dual structure.
Newton System: The method employs a reduced elastic KKT system solved via Newton's method. It incorporates proximal penalties on primal and dual variables to ensure robustness under degeneracy.
Convergence: The solver is guaranteed to converge to the "closest-to-feasible" solution (in a least-violation $\ell_2$ sense) when no feasible point exists.

2. Dual Variable Refinement

While the elastic formulation provides a robust primal solution, the associated dual variables encode penalized violations rather than true physical sensitivities (e.g., contact forces). To address this, the authors introduce a lightweight refinement stage. This stage warm-starts from the elastic primal solution, shifts the constraints by the optimal elastic relaxation variables, and re-solves the hard-constrained problem. This recovers physically meaningful dual variables and consistent sensitivities in just a few iterations.

3. Differentiable and SQP Extensions

ELASTIC ODYNLAYER: A differentiable QP layer that extends the solver to learning pipelines. It remains strictly convex and $C^1$ (continuously differentiable) even under infeasibility. The backward pass combines implicit gradients through the KKT system with explicit corrections derived from the elastic relaxation variables, ensuring stable gradients for parameter identification even when constraints conflict.
ELASTIC ODYNSQP: An infeasibility-aware SQP method for nonlinear optimal control. It resolves inconsistent linearizations and intrinsically infeasible tasks by selectively relaxing specific constraint categories (e.g., dynamics vs. kinematics) into the elastic block while keeping others hard. It employs a merit function that balances elastic penalties and hard residuals to guarantee descent.

Key Contributions

The paper outlines four primary contributions:

ELASTIC ODYN Solver: A smooth $\ell_2$ -elastic non-interior-point QP solver capable of robustly handling infeasible, degenerate, and ill-conditioned problems.
Dual Recovery: A refinement procedure that recovers interpretable dual variables from the elastic solution, making the framework suitable for contact-physics applications.
Differentiable Optimization: ELASTIC ODYNLAYER, a projection-free, fully smooth QP layer that yields well-defined derivatives for both feasible and infeasible problems, enabling gradient-based learning in constrained settings.
Infeasibility-Aware SQP: ELASTIC ODYNSQP, a framework that resolves inconsistent subproblems and infeasible optimal control tasks through selective constraint relaxation.

Experimental Results

The framework was evaluated across four domains:

Benchmark QPs: On the Maros–Mészáros test set, ELASTIC ODYN demonstrated superior warm-start performance and lower failure rates compared to state-of-the-art solvers like PROXQP, OSQP, and GUROBI, particularly in high-accuracy and perturbed regimes.
Ill-Conditioned/Infeasible QPs: In tests with deliberately conflicting constraints and poor scaling, ELASTIC ODYN consistently converged to the analytical least-squares solutions, whereas competitors often failed or timed out.
Contact Mechanics: In a frictionless disc-in-tube simulation involving degenerate contact normals, the solver successfully resolved singularities by admitting controlled penetration, restoring force generation where standard methods would fail.
Learning and Control:
- Parameter Identification: The differentiable layer successfully trained a restitution coefficient in an elastic impact setting, converging smoothly to the ground truth despite initial infeasibility.
- Trajectory Optimization: In quadrupedal (ANYmal) and humanoid (Talos) locomotion tasks, ELASTIC ODYNSQP converged to feasible or least-squares solutions where hard-constrained SQP methods stalled or diverged, even under conflicting targets and tight constraints.

Significance

The paper claims that ELASTIC ODYN unifies infeasibility handling, numerical robustness, and differentiability, addressing a fundamental deficiency in existing optimization-based robotics and AI pipelines. By moving beyond strict feasibility assumptions, the framework enables:

Robust Control: Reliable operation in dynamic environments where constraints may temporarily conflict.
Stable Learning: The ability to train optimization layers through infeasible regions without gradient collapse.
Unified Framework: A single approach that handles simulation, control, and learning tasks that were previously prone to solver breakdown due to infeasibility or degeneracy.

The authors position this work as a step toward optimization-based robotics that can operate effectively "beyond the feasibility assumptions of existing methods."