发表机构
Waabi Innovation Inc.（瓦比创新公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出平方根Gauss-Newton iLQR，利用代价函数的加权最小二乘结构，使反向传播每步仅需一次QR分解，直接提取反馈增益与Cholesky因子，显著提升数值稳定性。
AI 中文摘要
迭代线性二次调节器（iLQR）是一种广泛用于非线性轨迹优化的算法。在每次迭代中，它通过动态规划求解问题的局部线性二次近似，并传播一个二次代价函数。如果代价函数近似的Hessian矩阵是半正定的，则可以推导出iLQR的平方根形式，该形式转而传播其Cholesky因子。这带来了显著的数值优势——正如平方根卡尔曼滤波器相比其传统对应物的改进——尤其是在iLQR被用于增广拉格朗日框架处理约束时，其中大的惩罚项会恶化条件数。先前存在iLQR及相关算法的平方根形式，但它们要么在数值上次优，要么在算法上复杂，或两者兼有。在本文中，我们表明有效的平方根形式的关键在于代价函数的Gauss-Newton（加权最小二乘）结构：这产生了一个半正定性，该性质从Hessian矩阵扩展到整个增广代价矩阵，并使得一个极其简单的反向传播成为可能，其中每一步简化为一次QR分解，从该分解中直接提取反馈增益和传播的Cholesky因子。
英文摘要
The iterative Linear Quadratic Regulator (iLQR) is a widely used algorithm for nonlinear trajectory optimization. At each iteration, it solves a local linear-quadratic approximation of the problem via dynamic programming, propagating a quadratic cost-to-go function. If the Hessian of the cost-to-go approximation is positive-semidefinite, one can derive a square root formulation of iLQR that propagates its Cholesky factor instead. This offers significant numerical advantages - much as square root Kalman filters improve upon their conventional counterparts - particularly when iLQR is used within an augmented Lagrangian framework for handling constraints, where large penalties degrade conditioning. Previous square root formulations of iLQR and related algorithms exist, but they are either numerically suboptimal, algorithmically complex, or both. In this paper, we show that the key to an effective square root formulation lies in the Gauss-Newton (weighted least-squares) structure of the cost function: this yields a positive semidefiniteness property that extends beyond the Hessian to the full augmented cost-to-go matrix, and enables a backward pass of remarkable simplicity in which each step reduces to a single QR-decomposition, from which the feedback gain and propagated Cholesky factor are extracted directly.
Comments16 pages, 2 figures, ISRR 2026
