用代数方法高效规划并验证3自由度机械臂轨迹
Trajectory Planning and Certification for 3-DOF Robot Manipulators Using Real Quantifier Elimination Based on Comprehensive Gröbner Systems
- 基于综合格罗滕迪克基的量化消除法，参数化求解逆运动学
- 可在端点轨迹任意点验证解的存在性，支持直线与三次样条轨迹
- 在Risa/Asir系统中实现，兼顾效率与数学严格性
本文提出一种针对3自由度机械臂的轨迹规划与认证算法及其实现。该方法基于基于综合格罗滕迪克基（CGS）的实量化消除（CGS-QE）。其核心优势在于轨迹规划与解认证的高效性，源于对CGS的有效利用。首先，在轨迹规划中，通过格罗滕迪克基计算求解轨迹上每一点的逆运动学问题；传统方法需逐点重算格罗滕迪克基，耗时较长，本文通过为参数化系统计算CGS，将末端执行器坐标作为参数，避免重复计算，显著提升效率。其次，在解认证方面，该方法可证明在末端执行器轨迹任意点均存在逆运动学解，并适用于由线段和三次自然样条组成的轨迹。算法在计算机代数系统Risa/Asir中实现。
原文摘要 · Abstract (English)
We propose an algorithm and its implementation for trajectory planning and certification for 3-DOF robot manipulators. The method uses Real Quantifier Elimination (QE) based on Comprehensive Gröbner Systems (CGS), also known as the CGS-QE method. The main advantage of the proposed method is its efficiency in trajectory planning and solution certification. This efficiency comes from the effective use of the CGS. First, for trajectory planning, we solve the inverse kinematics problem at each point along the trajectory via Gröbner basis computation. This usually requires recalculating the Gröbner basis at every point, which is time-consuming. We avoid this by computing the CGS for a parametric system. Here, the end-effector coordinates are parameters. This approach streamlines the algorithm. Second, for solution certification, the CGS-QE method certifies that an inverse kinematics solution exists at any point along the end-effector's trajectory. Our method also certifies solutions for trajectories composed of line segments and cubic natural splines. The algorithm is implemented within the computer algebra system Risa/Asir.
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