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.

本文提出了基于距离几何的逆运动学求解器，并结合 Riemannian 优化算法，初值平滑技术等方法，实现了更高效的求解方法。