Trajectory Planning and Certification for 3-DOF Robot Manipulators Using Real Quantifier Elimination Based on Comprehensive Gröbner Systems

Jul 16, 2026

An algorithm and implementation enable trajectory planning and rigorous solution certification for 3-DOF robot manipulators. The approach uses Real Quantifier Elimination (QE) with Comprehensive Gröbner Systems (CGS), or the CGS-QE method, to improve efficiency. For planning, inverse kinematics is solved along the path via Gröbner basis computation without recomputing bases at each point by precomputing CGS for a parametric system. For certification, CGS-QE guarantees inverse kinematics solution existence for every point, including line segments and cubic natural splines, implemented in Risa/Asir.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages9

Reading Time23 min

Introduction

Motion planning and inverse kinematics

CGS-based CGS-QE for certification

Inverse kinematics of myCobot as a 3-DOF manipulator

Problem formulation

Trajectory planning for myCobot

Line segments and cubic natural splines

Solution certification using CGS-QE

arXiv :2607 . 11657v1 [ cs .RO] 13 Jul 2026 Trajectory Planning and Certification for 3-DOF Robot Manipulators Using Real Quantifier Elimination Based on Comprehensive Gr¨obner Systems Yu Nakai 1 , Akira Terui 1 [0000-0003-0846-3643], and Masahiko Mikawa 1 [0000-0002-2193-3198] University of Tsukuba, Tsukuba, Japan [terui@math.tsukuba.ac.jp](terui@math.tsukuba.ac.jp) [mikawa@slis.tsukuba.ac.jp](mikawa@slis.tsukuba.ac.jp) [https://researchmap.jp/aterui](https://researchmap.jp/aterui) Abstract. 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 Gr¨obner 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 Gr¨obner basis computation. This usually requires recalculating the Gr¨obner 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, and experimental results are presented. Keywords: Comprehensive Gr¨obner Systems · Robotics · Inverse kinematics · Trajectory planning 1 Introduction In this paper, we discuss the motion planning of a robot manipulator. A manipulator is a robot that consists of a series of links connected by joints like a human arm. The last link connected to the end-effector. We deal with a manipulator “myCobot 280” [3] (hereafter, we refer to it as “myCobot”), which has six revolute joints and therefore is a 6-DOF manipulator. 2 Yu Nakai, Akira Terui, and Masahiko Mikawa Motion planning involves solving the inverse kinematics and trajectory planning problems. Inverse kinematics finds a joint configuration for a given endeffector position and orientation. Trajectory planning extends this problem to a continuous end-effector path. In computer algebra, the inverse kinematics problem is solved by reducing it to a system of polynomial equations and then computing a Gr¨obner basis (see [4,6,12] and the references therein) . The trajectory planning problem is solved by applying inverse kinematics at each point along the trajectory. An advantage of using Gr¨obner bases is that we can find global solutions, which allows the feasibility of the end-effector’s motion to be determined before actual execution, unlike numerical methods. However, Gr¨obner basis computation can be more computationally intensive than numerical methods, and repeated computation along the trajectory can be time-consuming. Our research group has proposed methods for solving inverse kinematics and trajectory planning problems for manipulators using Gr¨obner bases [8–11,15] . A core feature of our proposed method is the use of Comprehensive Gr¨obner Systems (CGS) [13,14] to avoid the repeated computation of Gr¨obner bases by precomputing the CGS for a parametric system of equations, thereby significantly reducing computational cost. In this paper, we focus on the CGS-based Quantifier Elimination (CGSQE) method [5], another application of CGS, for certifying solutions to trajectory planning problems. Our research group’s previous studies applied CGS-QE to inverse kinematics and trajectory planning for 3-DOF manipulators [10,15], guaranteeing solution existence along a trajectory. A core feature of our method is that, by representing points o

FAQs

What is the main method proposed for trajectory planning and certification?

The method uses Real Quantifier Elimination based on Comprehensive Gröbner Systems (CGS), known as the CGS-QE approach.

How does the approach improve efficiency during trajectory planning?

It computes CGS for a parametric system so that Gröbner basis computation does not need to be repeated at every trajectory point.

What does the certification guarantee for the trajectory?

It certifies that an inverse kinematics solution exists at any point along the end-effector trajectory.

Jul 16, 2026

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