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Enumeration of spatial manipulators by using the concept of Adjacency Matrix

Akkarapakam Suneesh Jacob, Bhaskar Dasgupta, Rituparna Datta

TL;DR

This work tackles the problem of enumerating spatial manipulators by extending the adjacency-matrix representation from planar to 3D mechanisms while explicitly distinguishing base and end-effector links. It adopts a brute-force generation of all possible $n\times n$ adjacency matrices using four joint types $\{R,P,C,S\}$ plus nonconnection, then filters candidates with a sequence of validity and nonisomorphism criteria, including the DOF condition $D=6(n-1)-5(n_r+n_p)-4n_c-3n_s$. The study reports counts of unique manipulators: 96 for 1-DOF, 700 for 2-DOF, 8 for 3-DOF, and 15 for 4-DOF across 4- to 5-link topologies, organized into classes, with schematic references. It provides an atlas of spatial manipulator topologies to support dimensional synthesis in a companion study, while acknowledging limitations of the Kutzbach criterion and the need for broader enumeration in future work, including more joints and link counts.

Abstract

This study is on the enumeration of spatial robotic manipulators, which is an essential basis for a companion study on dimensional synthesis, both of which together present a wider utility in manipulator synthesis. The enumeration of manipulators is done by using adjacency matrix concept. In this paper, a novel way of applying adjacency matrix to spatial manipulators with four types of joints, namely revolute, prismatic, cylindrical and spherical joints, is presented. The limitations of the applicability of the concept to 3D manipulators are discussed. 1-DOF (Degree Of Freedom) manipulators of four links and 2-DOF, 3-DOF and 4-DOF manipulators of three links, four links and five links, are enumerated based on a set of conventions and some assumptions. Finally, 96 1-DOF manipulators of four links, 641 2-DOF manipulators of 5 links, 4 2-DOF manipulators of three links, 8 3-DOF manipulators of four links and 15 4-DOF manipulators of five links are presented.

Enumeration of spatial manipulators by using the concept of Adjacency Matrix

TL;DR

This work tackles the problem of enumerating spatial manipulators by extending the adjacency-matrix representation from planar to 3D mechanisms while explicitly distinguishing base and end-effector links. It adopts a brute-force generation of all possible adjacency matrices using four joint types plus nonconnection, then filters candidates with a sequence of validity and nonisomorphism criteria, including the DOF condition . The study reports counts of unique manipulators: 96 for 1-DOF, 700 for 2-DOF, 8 for 3-DOF, and 15 for 4-DOF across 4- to 5-link topologies, organized into classes, with schematic references. It provides an atlas of spatial manipulator topologies to support dimensional synthesis in a companion study, while acknowledging limitations of the Kutzbach criterion and the need for broader enumeration in future work, including more joints and link counts.

Abstract

This study is on the enumeration of spatial robotic manipulators, which is an essential basis for a companion study on dimensional synthesis, both of which together present a wider utility in manipulator synthesis. The enumeration of manipulators is done by using adjacency matrix concept. In this paper, a novel way of applying adjacency matrix to spatial manipulators with four types of joints, namely revolute, prismatic, cylindrical and spherical joints, is presented. The limitations of the applicability of the concept to 3D manipulators are discussed. 1-DOF (Degree Of Freedom) manipulators of four links and 2-DOF, 3-DOF and 4-DOF manipulators of three links, four links and five links, are enumerated based on a set of conventions and some assumptions. Finally, 96 1-DOF manipulators of four links, 641 2-DOF manipulators of 5 links, 4 2-DOF manipulators of three links, 8 3-DOF manipulators of four links and 15 4-DOF manipulators of five links are presented.
Paper Structure (18 sections, 19 equations, 8 figures, 5 tables, 6 algorithms)

This paper contains 18 sections, 19 equations, 8 figures, 5 tables, 6 algorithms.

Table of Contents

  1. Introduction and Literature Review
  2. Analysis and Methodology
  3. Adjacency matrix representation
  4. On the applicability of adjacency matrix representation of spatial manipulators
  5. Methodology to enumerate manipulators
  6. Algorithms
  7. Conditions 1 and 2
  8. Condition 3
  9. Conditions 4, 5, 6, 7 and 8
  10. Conditions 9, 10, 11 and 12
  11. Isomorphism detection and elimination
  12. Results
  13. Enumeration of manipulators of DOF 1
  14. Enumeration of manipulators of DOF 2
  15. Enumeration of manipulators of DOF 3
  16. ...and 3 more sections

Figures (8)

  • Figure 1: Typical adjacency matrix structure
  • Figure 2: Manipulator with a non-contributing open-chain
  • Figure 3: Manipulator with a non-contributing loop
  • Figure 4: Manipulator having two actuators and one independent component of end-effector velocity
  • Figure 5: Schematic diagram of a manipulator having less than three joints of revolute motion in a loop
  • ...and 3 more figures