距离几何逆运动学的凸迭代
Convex Iteration for Distance-Geometric Inverse Kinematics
摘要：
逆运动学（IK）是寻找满足一个或多个末端执行器的位置或姿态约束的机器人关节配置的问题。对于具有冗余自由度的机器人，通常存在无穷多的非凸解集。当工作空间中的障碍物施加碰撞避免约束时，IK问题进一步复杂化。在一般情况下，封闭形式的表达式产生可行的配置不存在，激励使用数值求解方法。然而，这些方法依赖于非凸问题的局部优化，通常需要精确的初始化或多次重新初始化才能收敛到有效的解。在这项工作中，我们首先制定逆运动学复杂的工作空间约束作为一个凸可行性问题，其低秩可行点提供精确的IK解决方案。然后，我们提出了\texttt{CIDGIK}（凸迭代距离几何逆运动学），一个算法，解决了这个可行性问题的一系列半定方案，其目标是鼓励低秩极小。我们的问题制定优雅地统一了机器人的配置空间和工作空间的约束：内在的机器人几何形状和避障都表示为简单的线性矩阵方程和不等式。我们对各种流行的机械手模型的实验结果表明，更快，更准确的收敛比传统的非线性优化为基础的方法，特别是在环境中有许多障碍物。
英文摘要：
Inverse kinematics (IK) is the problem of finding robot joint configurations that satisfy constraints on the position or pose of one or more end-effectors. For robots with redundant degrees of freedom, there is often an infinite, nonconvex set of solutions. The IK problem is further complicated when collision avoidance constraints are imposed by obstacles in the workspace. In general, closed-form expressions yielding feasible configurations do not exist, motivating the use of numerical solution methods. However, these approaches rely on local optimization of nonconvex problems, often requiring an accurate initialization or numerous re-initializations to converge to a valid solution. In this work, we first formulate inverse kinematics with complex workspace constraints as a convex feasibility problem whose low-rank feasible points provide exact IK solutions. We then present \texttt{CIDGIK} (Convex Iteration for Distance-Geometric Inverse Kinematics), an algorithm that solves this feasibility problem with a sequence of semidefinite programs whose objectives are designed to encourage low-rank minimizers. Our problem formulation elegantly unifies the configuration space and workspace constraints of a robot: intrinsic robot geometry and obstacle avoidance are both expressed as simple linear matrix equations and inequalities. Our experimental results for a variety of popular manipulator models demonstrate faster and more accurate convergence than a conventional nonlinear optimization-based approach, especially in environments with many obstacles.
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