Denavit-Hartenberg Parameters

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A commonly used convention for selecting frames of reference in robotics applications is the Denavit and Hartenberg (D-H) convention which introduced by Jaques Denavit and Richard S. Hartenberg. In this convention, each homogeneous transformation is represented as a product of four basic transformations. The common normal between two lines was the main geometric concept that allowed Denavit and Hartenberg to find a minimal representation. The transformation is described by the following four parameters known as D-H Parameters:[1]

  • a\,: length
  • \alpha\,: twist
  • d\,: offset
  • \theta\,: angle

Since only four parameters are used, the frames that can be represented this way has to satisfy two more constraints

  1. the xn-axis is perpendicular to the zn − 1 axis
  2. the xn-axis intersects zn − 1 axis


Every link/joint pair can be described as a coordinate transformation from the previous coordinate system to the next coordinate system.

{}^{n - 1}T_n = \operatorname{Trans}_{z_{n - 1}}(d_n) \cdot \operatorname{Rot}_{z_{n - 1}}(\theta_n) \cdot \operatorname{Trans}_{x_n}(a_n) \cdot \operatorname{Rot}_{x_n}(\alpha_n)

Note that these are 2 screws after oneanother. See Screw (motion).

The matrices mentioned above are as follows:

\operatorname{Trans}_{z_{n - 1}}(d_n) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & d_n \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}
\operatorname{Rot}_{z_{n - 1}}(\theta_n) = \begin{pmatrix} \cos\theta_n & -\sin\theta_n & 0 & 0 \\ \sin\theta_n & \cos\theta_n & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}
\operatorname{Trans}_{x_n}(a_n) = \begin{pmatrix} 1 & 0 & 0 & a_n \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}
\operatorname{Rot}_{x_n}(\alpha_n) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\alpha_n & -\sin\alpha_n & 0 \\ 0 & \sin\alpha_n & \cos\alpha_n & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}

This gives:

\operatorname{}^{n - 1}T_n = \begin{pmatrix} \cos\theta_n & -\sin\theta_n \cos\alpha_n & \sin\theta_n \sin\alpha_n & a_n \cos\theta_n \\ \sin\theta_n & \cos\theta_n \cos\alpha_n & -\cos\theta_n \sin\alpha_n & a_n \sin\theta_n \\ 0 & \sin\alpha_n & \cos\alpha_n & d_n \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}

[edit] References

  1. ^ Spong, M., M. Vidyasagar, ”Robot Dynamics and Control”, John Wiley and Sons, 1989
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