Gradient-based methods can efficiently optimize controllers using physical priors and differentiable simulators, but contact-rich manipulation remains challenging due to discontinuous or vanishing gradients from hybrid contact dynamics. Smoothing the dynamics yields continuous gradients, but the resulting model mismatch can cause controller failures when executed on real systems. We address this trade-off by planning with smoothed dynamics while explicitly quantifying and compensating for the induced errors, providing formal guarantees of constraint satisfaction and goal reachability on the true hybrid dynamics. Our method smooths both contact dynamics and geometry via a novel differentiable simulator based on convex optimization, which enables us to characterize the discrepancy from the true dynamics as a set-valued deviation. This deviation constrains the optimization of time-varying affine feedback policies through analytical bounds on the system's reachable set, enabling robust constraint satisfaction guarantees for the true closed-loop hybrid dynamics, while relying solely on informative gradients from the smoothed dynamics. We evaluate our method on several contact-rich tasks, including planar pushing, object rotation, and in-hand dexterous manipulation, achieving guaranteed constraint satisfaction with lower safety violation and goal error than baselines. By bridging differentiable physics with set-valued robust control, our method is the first certifiable gradient-based policy synthesis method for contact-rich manipulation.

翻译：梯度方法能够利用物理先验与可微仿真器高效优化控制器，但接触式操作因混合接触动力学产生的不连续或消失梯度而仍具挑战性。对动力学进行平滑处理可获得连续梯度，但由此产生的模型失配会在真实系统执行时导致控制器失效。我们通过采用平滑动力学进行规划，同时显式量化并补偿其引发的误差，从而解决这一权衡问题，为真实混合动力学提供约束满足与目标可达性的形式化保证。本方法通过基于凸优化的新型可微仿真器对接触动力学与几何结构同时进行平滑处理，使我们能够将真实动力学的偏差表征为集值偏离量。该偏离量通过系统可达集解析边界约束时变仿射反馈策略的优化，在仅依赖平滑动力学信息梯度的前提下，为真实闭环混合动力学提供鲁棒的约束满足保证。我们在多项接触式任务（包括平面推动、物体旋转及手内灵巧操作）上评估本方法，相比基线方法实现了更低的约束违反率与目标误差，同时确保约束满足的严格保证。通过融合可微物理与集值鲁棒控制，本方法成为首个适用于接触式操作的具备可认证性的梯度策略合成方法。