Inverse Optimal Control (IOC) aims to recover the cost function that explains observed trajectories as solutions of an optimal control problem. Classical IOC formulations rely on bilevel optimization, which repeatedly solves a nested optimal control problem and quickly becomes computationally prohibitive for realistic systems. Recent projection-based approaches offer a promising alternative but suffer from numerical instability when solved with gradient-based methods due to violations of standard constraint qualifications. In this paper, we show that these difficulties stem from the geometric structure of the IOC feasible set. We demonstrate that the set of trajectories satisfying the optimality conditions naturally forms a manifold and reformulate IOC as an optimization problem on this manifold. Based on this insight, we propose a Riemannian Inverse Optimal Control (RIOC) method that projects observed trajectories onto the manifold of optimal solutions while preserving feasibility by construction. Experiments on real human arm trajectories show that the proposed method achieves comparable or better reconstruction accuracy than classical bilevel IOC while reducing computation time by about a factor of four. These results highlight the potential of geometric optimization methods to improve the scalability and reliability of IOC for robotics and human motion analysis.

翻译：逆最优控制（IOC）旨在恢复能够解释观测轨迹作为最优控制问题解的成本函数。经典IOC公式依赖于双层优化，该方法反复求解嵌套的最优控制问题，对于实际系统而言很快就会变得计算上不可行。近年来基于投影的方法提供了一种有前景的替代方案，但在使用基于梯度的方法求解时，由于违反标准约束规范，会遭受数值不稳定性。在本文中，我们表明这些困难源于IOC可行集的几何结构。我们证明满足最优性条件的轨迹集自然构成一个流形，并将IOC重新表述为该流形上的优化问题。基于这一见解，我们提出了一种黎曼逆最优控制（RIOC）方法，该方法将观测轨迹投影到最优解流形上，同时通过构造保持可行性。在真实人类手臂轨迹上的实验表明，所提出的方法在重建精度上与经典双层IOC相当或更优，同时计算时间减少了约四倍。这些结果凸显了几何优化方法在提升IOC用于机器人学和人体运动分析的可扩展性和可靠性方面的潜力。