Solving the inverse kinematics problem is a fundamental challenge in motion planning, control, and calibration for articulated robots. Kinematic models for these robots are typically parametrized by joint angles, generating a complicated mapping between the robot configuration and the end-effector pose. Alternatively, the kinematic model and task constraints can be represented using invariant distances between points attached to the robot. In this paper, we formalize the equivalence of distance-based inverse kinematics and the distance geometry problem for a large class of articulated robots and task constraints. Unlike previous approaches, we use the connection between distance geometry and low-rank matrix completion to find inverse kinematics solutions by completing a partial Euclidean distance matrix through local optimization. Furthermore, we parametrize the space of Euclidean distance matrices with the Riemannian manifold of fixed-rank Gram matrices, allowing us to leverage a variety of mature Riemannian optimization methods. Finally, we show that bound smoothing can be used to generate informed initializations without significant computational overhead, improving convergence. We demonstrate that our inverse kinematics solver achieves higher success rates than traditional techniques, and substantially outperforms them on problems that involve many workspace constraints.

翻译：求解逆运动学问题是铰接式机器人运动规划、控制与标定中的一项基础性挑战。此类机器人的运动学模型通常以关节角参数化，从而在机器人构型与末端执行器位姿之间产生复杂映射关系。另一种思路是利用附着于机器人本体上的点间不变距离来表示运动学模型与任务约束。本文针对一大类铰接式机器人及任务约束，正式论证了基于距离的逆运动学与距离几何问题之间的等价性。不同于以往方法，我们利用距离几何与低秩矩阵补全的内在关联，通过局部优化完成部分欧氏距离矩阵，从而求解逆运动学问题。进一步地，我们将欧氏距离矩阵空间参数化为固定秩格兰姆矩阵的黎曼流形，使得多种成熟的黎曼优化方法得以应用。最后，我们证明无显著计算开销的边界平滑处理可生成优化初始化值，从而提升收敛性能。实验表明，我们的逆运动学求解器在成功率上超越传统技术，并在涉及多工作空间约束的问题中展现出显著优势。