用光滑样条与代数方法优化机械臂路径，提升运动效率与精度。
An Optimized Path Planning of Manipulator Using Spline Curves and Real Quantifier Elimination Based on Comprehensive Gröbner Systems
- 结合三次样条插值生成平滑轨迹，提升路径连续性。
- 基于综合格罗布纳系的量化消去法高效求解逆运动学并验证解的存在性。
- 通过最短路径算法优化关节配置，减少总运动量，适合高精度轨迹规划场景。
本文提出一种改进的机械臂逆运动学与最优路径规划方法。逆运动学旨在确定末端执行器在给定位置和姿态下的关节角度；路径规划则寻求两点间的轨迹。传统计算机代数方法依赖格罗布纳基计算，虽能全局求解但计算成本高。本文采用综合格罗布纳系统（CGS）及其量化消去（CGS-QE）方法，高效求解逆运动学并验证轨迹规划解的存在性。进一步引入三次样条插值生成平滑路径，并利用最短路径算法优化关节配置，以最小化轨迹上各关节运动量之和。该方法显著增强了机械臂在复杂路径下的运动能力与序列优化性能。
原文摘要 · Abstract (English)
This paper presents an advanced method for addressing the inverse kinematics and optimal path planning challenges in robot manipulators. The inverse kinematics problem involves determining the joint angles for a given position and orientation of the end-effector. Furthermore, the path planning problem seeks a trajectory between two points. Traditional approaches in computer algebra have utilized Gröbner basis computations to solve these problems, offering a global solution but at a high computational cost. To overcome the issue, the present authors have proposed a novel approach that employs the Comprehensive Gröbner System (CGS) and CGS-based quantifier elimination (CGS-QE) methods to efficiently solve the inverse kinematics problem and certify the existence of solutions for trajectory planning. This paper extends these methods by incorporating smooth curves via cubic spline interpolation for path planning and optimizing joint configurations using shortest path algorithms to minimize the sum of joint configurations along a trajectory. This approach significantly enhances the manipulator's ability to navigate complex paths and optimize movement sequences.
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