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Geometric analysis of generic 3R robots, and necessary and sufficient conditions for a class of orthogonal robots to have four IKS
Durgesh Haribhau Salunkhe, Abhilash Nayak
TL;DR
Generic 3R robot analysis lacks a unified treatment because algebraic approaches are restricted to special architectures. The paper combines a conformal-geometric-algebra-derived inverse model with simplified algebraic analysis, establishing a framework for four-IKS conditions and necessary and sufficient results for a class of orthogonal robots.
Problem
Existing algebraic analyses of generic 3R robots are limited to special classes or architectural simplifications, while geometric analyses provide broader generic insight.
Method
The paper combines a geometric approach based on torus-geometric manifolds with simplified algebraic analysis of circle and torus intersections.
Results
The approach provides necessary and sufficient four-IKS conditions for orthogonal robots with d2 = 0 and d3 ≠ 0, including limits on z for transitions from four to two IKS.
Takeaways & Limitations
The framework links workspace and joint space while extending four-IKS analysis to an orthogonal-robot class not covered by the previously known conditions.
Takeaways & Limitations
The paper solves the orthogonal-robot case with d3 ≠ 0, while conditions for generic 3R robots remain ongoing.
Abstract
from arXiv · showhide
The kinematic analysis of a generic 3R robot has been investigated with multiple approaches in the past. The algebraic approaches have established concrete results but are unfortunately limited to special classes or architectural simplifications. Geometric approaches on the other hand have extended the analysis to generic robots while also providing an intuitive understanding of their kinematic properties. We use the best of both approaches to present the kinematic analysis of a generic 3R robot, using the inverse kinematic model inherited from a method based on conformal geometric algebra. The paper discusses a generic framework to study the conditions for a 3R robot to have four inverse kinematic solutions (IKS) and allows to study the distribution of IKS as seen in workspace. The necessary and sufficient conditions for a class of orthogonal robots are presented using the proposed approach.
1 Introduction
Prior analyses of generic positional 3R robots combine geometric and algebraic perspectives, but algebraic results remain limited to special architectures. This paper uses torus-geometric manifolds to analyze four-IKS conditions for generic robots and extends orthogonal-robot results.
- Generic positional 3R robots have been analyzed through geometric, computer-algebra, and algebraic approaches since Pieper’s 1968 work.
- Algebraic analyses provide concrete results but have focused on special classes or architectural simplifications, whereas geometric approaches extend to generic robots.
- Torus-geometric manifolds link workspace and joint space, enabling analysis of when a generic 3R robot has four inverse kinematic solutions.
- Parameterizing the torus by the last two joint axes allows joint-space singularities and inverse-solution distributions to be examined in workspace.
- The paper gives necessary and sufficient four-IKS conditions for orthogonal robots with d2 = 0 and d3 ≠ 0, extending earlier results for d2 ≠ 0 and d3 = 0.
2 Kinematic analysis of generic 3R robots
The paper reduces generic positional 3R inverse kinematics to intersections between a fixed circle and a rotating circle, then studies their torus geometry to classify robots by inverse-solution count. For orthogonal robots with d2 = 0, tangency and algebraic sign conditions yield necessary and sufficient quaternary conditions across geometric cases.
- A positional 3R robot is minimally represented by Denavit–Hartenberg parameters defining its architecture.
- Given an end-effector position, inverse kinematics reduces to intersecting a fixed circle CB with a rotating circle CA.The rotating circle traces a torus TA, whose intersection with CB can be examined in the plane z = zc.
- Four inverse kinematic solutions require four real intersections between the toric section TAz=zc and CB.Tangential circles identify limiting radii, after which a slight radius change can produce two additional nearby intersections.
- Orthogonal 3R robots: For d2 = 0, the torus is symmetric about the x-axis, and its tangency analysis simplifies because the slope equation factors into a circle and the line y = 0.Substituting y = 0 yields four solutions for the radii of the outermost and innermost tangential circles.
- Orthogonal 3R robots: The case r < R < a1 cannot be quaternary, while a1 = R is the only spindle-torus condition that prevents four solutions and also produces infinite-IKS degeneracy.The a1 = R example is shown as a binary 3R robot.
3 Conclusions
The paper combines geometric and simplified algebraic analysis to study four inverse kinematic solutions in generic 3R robots. It solves the orthogonal case with d3 ≠ 0, while generic-robot conditions remain ongoing.
- The paper uses a geometric approach with simplified algebraic analysis to study when generic 3R robots have four IKS.
- The approach already solves the conditions for orthogonal robots with d3 ≠ 0.
- The study of four-IKS conditions for generic 3R robots remains ongoing.