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:解决反动力学问题是运动规划、 控制和校准分解机器人的根本挑战。 这些机器人的数学模型通常通过联合角度进行对称, 从而在机器人配置和终端效应构成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学的等值, 以及大量分解机器人和任务限制的距离几何学问题。 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的 Euclidean 距离矩阵。 此外, 我们用 Riemann 的固定级格子矩阵将Euclidean 距离矩阵的空间进行对称, 使我们能够利用各种成熟的Riemannian 优化方法。 最后, 我们表明, 捆绑可以用来生成知情初始初始化的初始化, 而不显著的计算顶端, 改进合并 。 我们证明我们的传统空间限制率会大大超过 成功 。