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 parameterized by joint angles, generating a complicated mapping between a robot's configuration and 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 parameterize 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 novel inverse kinematics solver achieves higher success rates than traditional techniques, and significantly outperforms them on problems that involve many workspace constraints.
翻译:解决反动力学问题是运动规划、 控制和校准清晰机器人的基本挑战。 这些机器人的物理模型通常通过联合角度进行参数化, 从而在机器人配置和终端效应表面之间产生复杂的映射。 或者, 运动模型和任务限制可以使用与机器人相连接的点之间的不同距离来表示。 在本文中, 我们正式确定远程反动数学的等值, 以及对于一大批清晰的机器人和任务限制的远程几何问题。 与以往的做法不同, 我们使用距离几何和低级别矩阵完成之间的联系来寻找反动力学解决方案, 通过本地优化完成部分 Euclidean 远程矩阵。 此外, 我们将Euclidean 远程矩阵的空间与 Riemannian 固定级格拉姆矩阵的中位相匹配, 使我们能够利用各种成熟的 Riemannian 优化方法。 最后, 我们显示, 捆绑的平可以用来生成知情的初始初始的磁力限制率超过传统空间控制率。