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Geometric interpretation of the general POE model for a serial-link robot via conversion into D-H parameterization

Liao Wu, Ross Crawford, Jonathan Roberts

TL;DR

The paper develops an analytic method to convert a general POE kinematic model into a Denavit-Hartenberg (D-H) parameterization for serial-link robots with revolute, prismatic, and helical joints, enabling calibration workflows that transition between representations. It proves the equivalence of POE and DH for all three joint types and provides an explicit conversion framework based on four lemmas, including handling of base and tool frames. A key contribution is the identifiability analysis, showing the maximum identifiable parameter count as $C_3 = 5h+4r+2t+n+6$ and revealing that base/tool frame identifiability in the D-H view is restricted rather than arbitrary, with singular configurations highlighted. The approach is validated on the PUMA 560, demonstrating near-machine-precision equivalence between POE and DH forward kinematics and offering a practical calibration pathway that mitigates DH-based singularities by performing identification in POE space first.

Abstract

While Product of Exponentials (POE) formula has been gaining increasing popularity in modeling the kinematics of a serial-link robot, the Denavit-Hartenberg (D-H) notation is still the most widely used due to its intuitive and concise geometric interpretation of the robot. This paper has developed an analytical solution to automatically convert a POE model into a D-H model for a robot with revolute, prismatic, and helical joints, which are the complete set of three basic one degree of freedom lower pair joints for constructing a serial-link robot. The conversion algorithm developed can be used in applications such as calibration where it is necessary to convert the D-H model to the POE model for identification and then back to the D-H model for compensation. The equivalence of the two models proved in this paper also benefits the analysis of the identifiability of the kinematic parameters. It is found that the maximum number of identifiable parameters in a general POE model is 5h+4r +2t +n+6 where h, r, t, and n stand for the number of helical, revolute, prismatic, and general joints, respectively. It is also suggested that the identifiability of the base frame and the tool frame in the D-H model is restricted rather than the arbitrary six parameters as assumed previously.

Geometric interpretation of the general POE model for a serial-link robot via conversion into D-H parameterization

TL;DR

The paper develops an analytic method to convert a general POE kinematic model into a Denavit-Hartenberg (D-H) parameterization for serial-link robots with revolute, prismatic, and helical joints, enabling calibration workflows that transition between representations. It proves the equivalence of POE and DH for all three joint types and provides an explicit conversion framework based on four lemmas, including handling of base and tool frames. A key contribution is the identifiability analysis, showing the maximum identifiable parameter count as and revealing that base/tool frame identifiability in the D-H view is restricted rather than arbitrary, with singular configurations highlighted. The approach is validated on the PUMA 560, demonstrating near-machine-precision equivalence between POE and DH forward kinematics and offering a practical calibration pathway that mitigates DH-based singularities by performing identification in POE space first.

Abstract

While Product of Exponentials (POE) formula has been gaining increasing popularity in modeling the kinematics of a serial-link robot, the Denavit-Hartenberg (D-H) notation is still the most widely used due to its intuitive and concise geometric interpretation of the robot. This paper has developed an analytical solution to automatically convert a POE model into a D-H model for a robot with revolute, prismatic, and helical joints, which are the complete set of three basic one degree of freedom lower pair joints for constructing a serial-link robot. The conversion algorithm developed can be used in applications such as calibration where it is necessary to convert the D-H model to the POE model for identification and then back to the D-H model for compensation. The equivalence of the two models proved in this paper also benefits the analysis of the identifiability of the kinematic parameters. It is found that the maximum number of identifiable parameters in a general POE model is 5h+4r +2t +n+6 where h, r, t, and n stand for the number of helical, revolute, prismatic, and general joints, respectively. It is also suggested that the identifiability of the base frame and the tool frame in the D-H model is restricted rather than the arbitrary six parameters as assumed previously.

Paper Structure

This paper contains 12 sections, 4 theorems, 21 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The General POE Model
  3. Special Euclidean Group $SE(3)$
  4. General POE model for a serial-link robot
  5. D-H parameterization of the General POE model
  6. D-H parameterization
  7. Conversion from general POE Parameters into D-H Parameters
  8. Algorithm Validation
  9. Discussion on the Identifiability of Kinematic Parameters
  10. Identifiability of the POE Model
  11. Identifiability of the D-H Model
  12. Conclusion

Key Result

Lemma 1

For a helical joint, the exponential $e^{\bm{\hat{\bar{\xi}}}q}$ can be converted into the form of $\bm{H}R_Z(q)T_Z(hq)\bm{H^{-1}}$, where $h=\bar{\bm{\omega}}^T\bm{v}$ and $\bm{H}=R_Z(\theta)T_Z(d)R_X(\alpha)T_X(a)$.

Figures (4)

  • Figure 1: A general POE model for an n-DOF serial-link robot.
  • Figure 2: Standard D-H parameters for three basic 1-DOF lower pair joints.
  • Figure 3: D-H parameters for an n-joint serial-link robot with an arbitrarily located base frame and tool frame.
  • Figure 4: Rotational and translational errors between the normalized POE model and the converted D-H model with 100 randomly selected configurations.

Theorems & Definitions (5)

  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • Lemma 4