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An Optimized Path Planning of Manipulator Using Spline Curves and Real Quantifier Elimination Based on Comprehensive Gröbner Systems

Yusuke Shirato, Natsumi Oka, Akira Terui, Masahiko Mikawa

TL;DR

This paper addresses inverse kinematics and path planning for robot manipulators by integrating Comprehensive Gröbner Systems with QE (CGS-QE) and cubic spline interpolation to produce smooth trajectories. It extends prior CGS-based IK methods by enabling obstacle-aware spline paths and by optimizing joint motion along the trajectory via shortest-path strategies. The key contributions include a CGS-QE certified IK workflow, a spline-based trajectory generation algorithm, and two optimal-path selection approaches (greedy per-point and Dijkstra-based) with demonstrated practicality on a 3-DOF model of a manipulator. The findings indicate feasible computation times and improved joint-movement efficiency, suggesting potential for real-time deployment and applicability to other manipulators.

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 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.

An Optimized Path Planning of Manipulator Using Spline Curves and Real Quantifier Elimination Based on Comprehensive Gröbner Systems

TL;DR

This paper addresses inverse kinematics and path planning for robot manipulators by integrating Comprehensive Gröbner Systems with QE (CGS-QE) and cubic spline interpolation to produce smooth trajectories. It extends prior CGS-based IK methods by enabling obstacle-aware spline paths and by optimizing joint motion along the trajectory via shortest-path strategies. The key contributions include a CGS-QE certified IK workflow, a spline-based trajectory generation algorithm, and two optimal-path selection approaches (greedy per-point and Dijkstra-based) with demonstrated practicality on a 3-DOF model of a manipulator. The findings indicate feasible computation times and improved joint-movement efficiency, suggesting potential for real-time deployment and applicability to other manipulators.

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 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.

Paper Structure

This paper contains 12 sections, 15 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Solving the inverse kinematic problem
  3. Path and trajectory planning using spline interpolation
  4. Path planning using cubic spline interpolation
  5. Solving the inverse kinematics problem
  6. An algorithm for path and trajectory planning with spline interpolation
  7. Implementation and computational results
  8. Choosing an optimal path from multiple solutions of inverse kinematics
  9. Minimizing the magnitude of joint variations between each point
  10. Minimizing the sum of the magnitudes of the joint variations from the starting point to the endpoint
  11. Implementation and computational results
  12. Concluding remarks

Figures (2)

  • Figure 1: The model of myCobot.
  • Figure 2: Four possible configurations of the joints at each point on the path.