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

2026-06-15

AI Summary

This work addresses the challenge of infeasible quadratic programs (QPs) in robotic systems—arising from conflicting objectives, modeling errors, or degenerate contacts—which commonly cause numerical failure in existing differentiable QP solvers that assume feasibility. To overcome this, we propose Elastic ODYN, a primal-dual non-interior-point QP solver based on a smooth ℓ₂ elastic relaxation that converges to the closest feasible solution when constraints are unsatisfiable and recovers physically consistent dual variables via lightweight refinement. Our method enables, for the first time, stable differentiable optimization over infeasible QPs, supports warm-starting, and robustly handles degenerate scenarios. We introduce the differentiable Elastic OdynLayer and an infeasibility-aware sequential quadratic programming (SQP) framework, Elastic OdynSQP. Experiments on standard QPs, singular contacts, parameter identification, and trajectory optimization for quadrupedal and humanoid robots demonstrate significant improvements over prior approaches in robustness, warm-start performance, and convergence reliability.

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-$\ell_2$ 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.

Problem

• infeasible quadratic programming
• robotic control
• constraint relaxation
• differentiable optimization
• degenerate contact conditions

Innovation

• elastic relaxation
• differentiable optimization
• infeasible QP
• primal-dual solver
• trajectory optimization