Inverse kinematics (IK) is central to robot control and motion planning, yet its nonlinear kinematic mapping makes it inherently nonconvex and particularly challenging under complex constraints. We present IKSPARK (Inverse Kinematics using Semidefinite Programming And RanK minimization), an obstacle-aware IK solver for robots with diverse morphologies, including open and closed kinematic chains with spherical, revolute, and prismatic joints. Our formulation expresses IK as a semidefinite programming (SDP) problem with additional rank-1 constraints on symmetric matrices with fixed traces. IKSPARK first solves the relaxed SDP, whose infeasibility certifies infeasibility of the original IK problem, and then recovers a rank-1 solution using iterative rank-minimization methods with proven local convergence. Obstacle avoidance is handled through a convexified formulation of mixed-integer constraints. Extensive experiments show that IKSPARK computes highly accurate solutions across various kinematic structures and constrained environments without post-processing. In obstacle-rich settings, especially fixed workcell environments, IKSPARK achieves substantially higher success rates than traditional nonlinear optimization methods.

翻译：逆运动学（IK）是机器人控制与运动规划的核心问题，但其非线性运动学映射导致该问题具有内在非凸性，在复杂约束条件下求解尤为困难。本文提出IKSPARK（基于半定规划与秩最小化的逆运动学求解器），该障碍感知逆运动学求解器适用于多种形态的机器人，包括包含球形关节、旋转关节和移动关节的开链与闭链运动链。我们将逆运动学问题表述为半定规划（SDP）问题，并附加对称矩阵上具有固定迹的秩-1约束。IKSPARK首先求解松弛后的半定规划问题，其不可行性即证明原逆运动学问题不可行；随后通过具有局部收敛保证的迭代秩最小化方法恢复秩-1解。障碍规避通过混合整数约束的凸化公式实现。大量实验表明，IKSPARK无需后处理即可在各种运动学结构和约束环境中计算高精度解。在障碍密集场景（尤其是固定工作单元环境）中，IKSPARK的成功率显著高于传统非线性优化方法。