Differentiable contact kinematics are essential for gradient-based methods in robotics, yet the mapping from robot state to contact distance, location, and normal becomes non-smooth in degenerate configurations of shapes with zero or undefined curvature. We address this inherent limitation by selectively regularizing such geometries into strictly convex implicit representations, restoring uniqueness and smoothness of the contact map. Leveraging this geometric regularization, we develop iDCOL, an implicit differentiable collision detection and contact kinematics framework. iDCOL represents colliding bodies using strictly convex implicit surfaces and computes collision detection and contact kinematics by solving a fixed-size nonlinear system derived from a geometric scaling-based convex optimization formulation. By applying the Implicit Function Theorem to the resulting system residual, we derive analytical derivatives of the contact kinematic quantities. We develop a fast Newton-based solver for iDCOL and provide an open-source C++ implementation of the framework. The robustness of the approach is evaluated through extensive collision simulations and benchmarking, and applicability is demonstrated in gradient-based kinematic path planning and differentiable contact physics, including multi-body rigid collisions and a soft-robot interaction example.

翻译：可微接触运动学对于机器人学中的梯度方法至关重要，然而在曲率为零或未定义的几何构型退化情况下，从机器人状态到接触距离、位置及法向的映射会变得非光滑。我们通过选择性地将此类几何结构正则化为严格凸的隐式表示，解决了这一固有局限性，从而恢复了接触映射的唯一性与光滑性。利用这种几何正则化，我们开发了iDCOL——一个隐式可微碰撞检测与接触运动学框架。iDCOL采用严格凸隐式曲面表示碰撞体，并通过求解基于几何缩放凸优化公式导出的固定规模非线性系统，计算碰撞检测与接触运动学。通过对所得系统残差应用隐函数定理，我们推导出接触运动学量的解析导数。我们为iDCOL开发了基于牛顿法的快速求解器，并提供了该框架的开源C++实现。通过大量碰撞仿真与基准测试评估了该方法的鲁棒性，并在梯度运动学路径规划与可微接触物理（包括多刚体碰撞及软体机器人交互示例）中验证了其适用性。