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Analysis of a Cuspidal 6R Robot

Alexander Feeß, Martin Weiß

TL;DR

This work analyzes the kinematics of a cuspidal 6R manipulator inspired by the KUKA Transpressor, showing that such a robot can admit up to $16$ inverse kinematic solutions. It combines a reduction to the initial $3$R chain with both analytic IK for special poses (vertical $z_E$-axis and poses on the $z_1$-axis) and a numerical solver for the general case, producing up to $16$ IK solutions and identifying four solution classes. It further proves cuspidality for a parameter class by deriving an analytic estimate of the Jacobian determinant along a path between IKS and demonstrating a singularity-free transition within classes; this is complemented by a numerical demonstration of the same grouping structure. The results advance understanding of 6R cuspidal robots and offer practical implications for safe motion planning by enabling singularity-free transitions and potential workspace analytics, even when analytical IK for the entire robot is not available.

Abstract

We present a theoretical and numerical analysis of the kinematics for the "Transpressor", a cuspidal 6R robot. It admits up to 16 inverse kinematics solutions which are described geometrically. For special target poses, we provide the solutions analytically and present a simple numerical solver for the general case. Moreover, an analytical estimate of the Jacobian determinant on a path between two solutions proves cuspidality for a class of robots similar to the transpressor.

Analysis of a Cuspidal 6R Robot

TL;DR

This work analyzes the kinematics of a cuspidal 6R manipulator inspired by the KUKA Transpressor, showing that such a robot can admit up to inverse kinematic solutions. It combines a reduction to the initial R chain with both analytic IK for special poses (vertical -axis and poses on the -axis) and a numerical solver for the general case, producing up to IK solutions and identifying four solution classes. It further proves cuspidality for a parameter class by deriving an analytic estimate of the Jacobian determinant along a path between IKS and demonstrating a singularity-free transition within classes; this is complemented by a numerical demonstration of the same grouping structure. The results advance understanding of 6R cuspidal robots and offer practical implications for safe motion planning by enabling singularity-free transitions and potential workspace analytics, even when analytical IK for the entire robot is not available.

Abstract

We present a theoretical and numerical analysis of the kinematics for the "Transpressor", a cuspidal 6R robot. It admits up to 16 inverse kinematics solutions which are described geometrically. For special target poses, we provide the solutions analytically and present a simple numerical solver for the general case. Moreover, an analytical estimate of the Jacobian determinant on a path between two solutions proves cuspidality for a class of robots similar to the transpressor.
Paper Structure (10 sections, 4 equations, 4 figures, 1 table)

This paper contains 10 sections, 4 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Robot Model
  3. Inverse Kinematic Solutions
  4. Kinematic of the Initial $3$R-Chain
  5. General Approach
  6. Poses with Vertical $z_E$-Axis
  7. Poses on $z_1$-Axis
  8. Number of IKS
  9. Singularity-free Change of Solution
  10. Conclusion and Outlook

Figures (4)

  • Figure 1: a The KUKA Transpressor, b our simplified version in home position (the red indication of axis $4$ is shifted along $z_4$ for visibility), and c the table of DH-parameters
  • Figure 2: Some IKS for a pose with vertical $z_E$-axis with a elbow up/down configurations, wrist outwards, and b left/right configurations, elbow down
  • Figure 3: A pose with $4$ IKS, $2$ in each elbow configuration. Considering both shoulder configurations yields a total of $8$ IKS. For the solid blue configuration, we indicated the direction of the axes with red lines. Note that axis $5$ is tangential to the circle.
  • Figure 4: Two configurations to reach a pose on the $z$-axis: left configuration with elbow up and no wrist flip (blue), and right configuration with elbow up and wrist flipped (green)

Theorems & Definitions (1)

  • remark thmcounterremark