An Optimized Path Planning of Manipulator Using Spline Curves and Real Quantifier Elimination Based on Comprehensive Gr\"obner Systems

TL;DR

This paper introduces a novel, efficient method for robot manipulator path planning that combines spline curves with comprehensive Gr"obner systems and quantifier elimination to improve solution speed and trajectory optimization.

Contribution

It extends existing algebraic methods by integrating spline interpolation and shortest path optimization for more effective and efficient manipulator path planning.

Findings

Enhanced path planning accuracy with spline curves.

Reduced computational cost using CGS-based methods.

Optimized joint configurations for smoother trajectories.

Abstract

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 Gr\"obner 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 Gr\"obner 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.