Efficient planning of assembly motions is a long standing challenge in the field of robotics that has been primarily tackled with reinforcement learning and sampling-based methods by using extensive physics simulations. This paper proposes a sample-efficient robust optimal control approach for the determination of assembly motions, which requires significantly less physics simulation steps during planning through the efficient use of derivative information. To this end, a differentiable physics simulation is constructed that provides second-order analytic derivatives to the numerical solver and allows one to traverse seamlessly from informative derivatives to accurate contact simulation. The solution of the physics simulation problem is made differentiable by using smoothing inspired by interior-point methods applied to both the collision detection as well as the contact resolution problem. We propose a modified variant of an optimization-based formulation of collision detection formulated as a linear program and present an efficient implementation for the nominal evaluation and corresponding first- and second-order derivatives. Moreover, a multi-scenario-based trajectory optimization problem that ensures robustness with respect to sim-to-real mismatches is derived. The capability of the considered formulation is illustrated by results where over 99\% successful executions are achieved in real-world experiments. Thereby, we carefully investigate the effect of smooth approximations of the contact dynamics and robust modeling on the success rates. Furthermore, the method's capability is tested on different peg-in-hole problems in simulation to show the benefit of using exact Hessians over commonly used Hessian approximations.

翻译：高效规划装配运动是机器人学领域长期存在的挑战，主要通过强化学习和基于采样的方法，并借助大量物理仿真来解决。本文提出了一种样本高效的鲁棒最优控制方法，用于确定装配运动，该方法通过有效利用导数信息，在规划过程中所需的物理仿真步骤显著减少。为此，构建了一个可微物理仿真，为数值求解器提供二阶解析导数，并允许从信息丰富的导数平滑过渡到精确的接触仿真。物理仿真问题的解通过采用受内点法启发的平滑技术而变得可微，该技术同时应用于碰撞检测和接触解析问题。我们提出了一种基于优化的碰撞检测公式的改进变体，该公式表述为线性规划，并给出了标称评估及相应一阶和二阶导数的有效实现。此外，推导了一个基于多场景的轨迹优化问题，以确保对仿真到现实不匹配的鲁棒性。所提公式的能力通过实验结果得到验证，在真实世界实验中实现了超过99%的成功执行率。在此过程中，我们仔细研究了接触动力学的平滑近似和鲁棒建模对成功率的影响。此外，该方法的能力在仿真中的不同孔轴装配问题上进行了测试，以展示使用精确Hessian矩阵相对于常用Hessian近似的优势。